<?xml version="1.0" encoding="UTF-8"?>

<!--XBRL Document Created with the Workiva Platform-->
<!--Copyright 2025 Workiva-->
<!--r:ab6c8579-7074-414e-bea3-4998b454a976,g:d3d10bf2-e9ba-448e-a320-4662003f3abf-->
<link:linkbase xmlns:link="http://www.xbrl.org/2003/linkbase" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.xbrl.org/2003/linkbase http://www.xbrl.org/2003/xbrl-linkbase-2003-12-31.xsd">
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/ConsolidatedBalanceSheets" xlink:type="simple" xlink:href="bhrb-20250630.xsd#ConsolidatedBalanceSheets"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/ConsolidatedBalanceSheets" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Deposits_2bd5bdb9-6c4c-4c2b-9bb6-bda8f9ecec3c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Deposits"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestBearingDepositLiabilities_8414b0e9-fafa-4b02-9dcd-819bf90cc328" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestBearingDepositLiabilities"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Deposits_2bd5bdb9-6c4c-4c2b-9bb6-bda8f9ecec3c" xlink:to="loc_us-gaap_InterestBearingDepositLiabilities_8414b0e9-fafa-4b02-9dcd-819bf90cc328" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoninterestBearingDepositLiabilities_a4bedfcc-140a-4876-a614-04441bf7e875" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoninterestBearingDepositLiabilities"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Deposits_2bd5bdb9-6c4c-4c2b-9bb6-bda8f9ecec3c" xlink:to="loc_us-gaap_NoninterestBearingDepositLiabilities_a4bedfcc-140a-4876-a614-04441bf7e875" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockholdersEquity_dce03c97-9b1e-4975-b210-dbce0e4c883c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockholdersEquity"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TreasuryStockCommonValue_c39477f2-8434-4c3f-b3ba-b298e4fe7460" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TreasuryStockCommonValue"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_StockholdersEquity_dce03c97-9b1e-4975-b210-dbce0e4c883c" xlink:to="loc_us-gaap_TreasuryStockCommonValue_c39477f2-8434-4c3f-b3ba-b298e4fe7460" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax_7550eecd-496a-4dba-a386-22944830ab05" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_StockholdersEquity_dce03c97-9b1e-4975-b210-dbce0e4c883c" xlink:to="loc_us-gaap_AccumulatedOtherComprehensiveIncomeLossNetOfTax_7550eecd-496a-4dba-a386-22944830ab05" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RetainedEarningsAccumulatedDeficit_11f00915-65fb-45d5-bc5f-82bcc4c2ac11" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RetainedEarningsAccumulatedDeficit"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_StockholdersEquity_dce03c97-9b1e-4975-b210-dbce0e4c883c" xlink:to="loc_us-gaap_RetainedEarningsAccumulatedDeficit_11f00915-65fb-45d5-bc5f-82bcc4c2ac11" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdditionalPaidInCapitalCommonStock_896817ea-b8f7-4ec7-8fe6-25d9d3a439b0" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdditionalPaidInCapitalCommonStock"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_StockholdersEquity_dce03c97-9b1e-4975-b210-dbce0e4c883c" xlink:to="loc_us-gaap_AdditionalPaidInCapitalCommonStock_896817ea-b8f7-4ec7-8fe6-25d9d3a439b0" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockValue_7c3ee647-4d37-4404-ac99-f5ebbdbe7e33" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockValue"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_StockholdersEquity_dce03c97-9b1e-4975-b210-dbce0e4c883c" xlink:to="loc_us-gaap_CommonStockValue_7c3ee647-4d37-4404-ac99-f5ebbdbe7e33" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredStockValue_52642298-0369-4975-af65-b576c48460af" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredStockValue"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_StockholdersEquity_dce03c97-9b1e-4975-b210-dbce0e4c883c" xlink:to="loc_us-gaap_PreferredStockValue_52642298-0369-4975-af65-b576c48460af" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_d902e35b-d315-4b69-bf66-c75af3a5f007" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableAllowanceForCreditLossExcludingAccruedInterest_74e373c8-fe64-4a29-b262-738b6ad64ea9" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableAllowanceForCreditLossExcludingAccruedInterest"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_d902e35b-d315-4b69-bf66-c75af3a5f007" xlink:to="loc_us-gaap_FinancingReceivableAllowanceForCreditLossExcludingAccruedInterest_74e373c8-fe64-4a29-b262-738b6ad64ea9" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_a2ba0079-2c89-4e38-9833-5eb92a31d159" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_d902e35b-d315-4b69-bf66-c75af3a5f007" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_a2ba0079-2c89-4e38-9833-5eb92a31d159" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_5ce7cec6-eef3-4ccd-948f-9cef58ed54ed" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashAndDueFromBanks_8217231a-802a-445f-af99-9dc0dc384a57" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashAndDueFromBanks"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_5ce7cec6-eef3-4ccd-948f-9cef58ed54ed" xlink:to="loc_us-gaap_CashAndDueFromBanks_8217231a-802a-445f-af99-9dc0dc384a57" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestBearingDepositsInBanks_05b66a76-1486-47b4-b07d-ea1ebb45743c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestBearingDepositsInBanks"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_5ce7cec6-eef3-4ccd-948f-9cef58ed54ed" xlink:to="loc_us-gaap_InterestBearingDepositsInBanks_05b66a76-1486-47b4-b07d-ea1ebb45743c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Liabilities_4bc55ac5-f17d-4bdb-8a50-1b3ba96f65e6" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Liabilities"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Deposits_7c8f9e3f-07f0-4112-84f3-48627bad1276" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Deposits"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Liabilities_4bc55ac5-f17d-4bdb-8a50-1b3ba96f65e6" xlink:to="loc_us-gaap_Deposits_7c8f9e3f-07f0-4112-84f3-48627bad1276" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShortTermBorrowings_05cd09f4-639c-412a-8e5c-80de8ef51e5d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShortTermBorrowings"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Liabilities_4bc55ac5-f17d-4bdb-8a50-1b3ba96f65e6" xlink:to="loc_us-gaap_ShortTermBorrowings_05cd09f4-639c-412a-8e5c-80de8ef51e5d" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SubordinatedDebt_31e8b2d8-5bcf-42db-874c-ed75b99dcfc4" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SubordinatedDebt"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Liabilities_4bc55ac5-f17d-4bdb-8a50-1b3ba96f65e6" xlink:to="loc_us-gaap_SubordinatedDebt_31e8b2d8-5bcf-42db-874c-ed75b99dcfc4" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_JuniorSubordinatedDebentureOwedToUnconsolidatedSubsidiaryTrust_cb0cbf99-cc24-44c7-8dfb-a3c196e35097" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_JuniorSubordinatedDebentureOwedToUnconsolidatedSubsidiaryTrust"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Liabilities_4bc55ac5-f17d-4bdb-8a50-1b3ba96f65e6" xlink:to="loc_us-gaap_JuniorSubordinatedDebentureOwedToUnconsolidatedSubsidiaryTrust_cb0cbf99-cc24-44c7-8dfb-a3c196e35097" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_InterestPayableAndOtherLiabilities_a8583459-50de-46d9-85e3-7b11b0241317" xlink:href="bhrb-20250630.xsd#bhrb_InterestPayableAndOtherLiabilities"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Liabilities_4bc55ac5-f17d-4bdb-8a50-1b3ba96f65e6" xlink:to="loc_bhrb_InterestPayableAndOtherLiabilities_a8583459-50de-46d9-85e3-7b11b0241317" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesAndStockholdersEquity_f1fa509e-1189-4e47-a8c2-16551b534c47" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesAndStockholdersEquity"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Liabilities_dfddf10b-448d-4e1e-961b-e23d8abc2c1c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Liabilities"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LiabilitiesAndStockholdersEquity_f1fa509e-1189-4e47-a8c2-16551b534c47" xlink:to="loc_us-gaap_Liabilities_dfddf10b-448d-4e1e-961b-e23d8abc2c1c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommitmentsAndContingencies_ce61c7f3-b6db-48c9-af9c-095c2ec28fee" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommitmentsAndContingencies"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LiabilitiesAndStockholdersEquity_f1fa509e-1189-4e47-a8c2-16551b534c47" xlink:to="loc_us-gaap_CommitmentsAndContingencies_ce61c7f3-b6db-48c9-af9c-095c2ec28fee" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockholdersEquity_7e27d270-7967-4f5f-bf98-5cd0238a0c5b" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockholdersEquity"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LiabilitiesAndStockholdersEquity_f1fa509e-1189-4e47-a8c2-16551b534c47" xlink:to="loc_us-gaap_StockholdersEquity_7e27d270-7967-4f5f-bf98-5cd0238a0c5b" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Assets"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherAssets_bc03e1a5-37bd-49f4-a3b9-04daaa3ab688" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherAssets"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_OtherAssets_bc03e1a5-37bd-49f4-a3b9-04daaa3ab688" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PropertyPlantAndEquipmentNet_dbc2827f-ce84-4ca4-b40a-3d6443f2d5ce" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PropertyPlantAndEquipmentNet"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_PropertyPlantAndEquipmentNet_dbc2827f-ce84-4ca4-b40a-3d6443f2d5ce" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LoansReceivableHeldForSaleAmount_1228b1d8-e90f-4749-beaa-c9f04c0fa25e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LoansReceivableHeldForSaleAmount"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_LoansReceivableHeldForSaleAmount_1228b1d8-e90f-4749-beaa-c9f04c0fa25e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_d575d28e-ecee-41d2-b074-ea53353d9350" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_d575d28e-ecee-41d2-b074-ea53353d9350" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherRealEstate_32ec9f70-0bb5-42dc-856b-8d26516bfd81" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherRealEstate"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_OtherRealEstate_32ec9f70-0bb5-42dc-856b-8d26516bfd81" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestReceivable_b88de57e-c4a2-44cc-aac3-72cf109a425a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestReceivable"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_InterestReceivable_b88de57e-c4a2-44cc-aac3-72cf109a425a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Goodwill_f63b5c66-02c4-47cc-84cc-9db8e5515eff" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Goodwill"/>
    <link:calculationArc order="7" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_Goodwill_f63b5c66-02c4-47cc-84cc-9db8e5515eff" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RestrictedInvestments_df1a88f2-9fae-461c-98c4-8eefe43e490e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RestrictedInvestments"/>
    <link:calculationArc order="8" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_RestrictedInvestments_df1a88f2-9fae-461c-98c4-8eefe43e490e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_5d26485c-f8b3-4a57-9345-c4efdd9ae341" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss"/>
    <link:calculationArc order="9" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_5d26485c-f8b3-4a57-9345-c4efdd9ae341" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_d64b6b35-7c3d-4ef9-bee9-11144826c047" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents"/>
    <link:calculationArc order="10" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_d64b6b35-7c3d-4ef9-bee9-11144826c047" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IntangibleAssetsNetExcludingGoodwill_ea6cdcf9-0616-42e2-844e-a51b304652bb" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IntangibleAssetsNetExcludingGoodwill"/>
    <link:calculationArc order="11" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_IntangibleAssetsNetExcludingGoodwill_ea6cdcf9-0616-42e2-844e-a51b304652bb" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BankOwnedLifeInsurance_62f3dbb9-cdbb-4007-854e-8b2629fbf713" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BankOwnedLifeInsurance"/>
    <link:calculationArc order="12" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_Assets_f090d363-d930-4dc0-a92f-d75b00efe27d" xlink:to="loc_us-gaap_BankOwnedLifeInsurance_62f3dbb9-cdbb-4007-854e-8b2629fbf713" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/ConsolidatedStatementsofIncome" xlink:type="simple" xlink:href="bhrb-20250630.xsd#ConsolidatedStatementsofIncome"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/ConsolidatedStatementsofIncome" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeExpenseNet_3e71bda6-4a54-4ade-a45d-bd835a536ccb" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeExpenseNet"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestExpenseOperating_11a88ba0-fa2c-449b-b704-e1c88cf52ce0" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestExpenseOperating"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeExpenseNet_3e71bda6-4a54-4ade-a45d-bd835a536ccb" xlink:to="loc_us-gaap_InterestExpenseOperating_11a88ba0-fa2c-449b-b704-e1c88cf52ce0" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeOperating_9eb811fb-36e8-47a6-9db2-6e7967c867fa" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeOperating"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeExpenseNet_3e71bda6-4a54-4ade-a45d-bd835a536ccb" xlink:to="loc_us-gaap_InterestIncomeOperating_9eb811fb-36e8-47a6-9db2-6e7967c867fa" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_c618f1bf-1d8d-4c7d-8d63-1c74c818787a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLoss_5f6adcd2-2bde-4920-8220-fa1b73521970" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLoss"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_c618f1bf-1d8d-4c7d-8d63-1c74c818787a" xlink:to="loc_us-gaap_NetIncomeLoss_5f6adcd2-2bde-4920-8220-fa1b73521970" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredStockDividendsIncomeStatementImpact_50a28a1f-fc92-4df7-8416-b45fa42c5b30" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredStockDividendsIncomeStatementImpact"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_c618f1bf-1d8d-4c7d-8d63-1c74c818787a" xlink:to="loc_us-gaap_PreferredStockDividendsIncomeStatementImpact_50a28a1f-fc92-4df7-8416-b45fa42c5b30" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoninterestIncome_6939bd84-fca8-4cd7-b64e-6de358c8bf19" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoninterestIncome"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtAndEquitySecuritiesGainLoss_8c05be44-2754-4979-a857-84ebfb7a8d22" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtAndEquitySecuritiesGainLoss"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestIncome_6939bd84-fca8-4cd7-b64e-6de358c8bf19" xlink:to="loc_us-gaap_DebtAndEquitySecuritiesGainLoss_8c05be44-2754-4979-a857-84ebfb7a8d22" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BankDebitAndOtherCardRevenue_9e2f7190-db8a-4191-9ca1-13997e1cec57" xlink:href="bhrb-20250630.xsd#bhrb_BankDebitAndOtherCardRevenue"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestIncome_6939bd84-fca8-4cd7-b64e-6de358c8bf19" xlink:to="loc_bhrb_BankDebitAndOtherCardRevenue_9e2f7190-db8a-4191-9ca1-13997e1cec57" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoninterestIncomeOtherOperatingIncome_291c0a8f-9d96-4fa1-ad28-308f18f34057" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoninterestIncomeOtherOperatingIncome"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestIncome_6939bd84-fca8-4cd7-b64e-6de358c8bf19" xlink:to="loc_us-gaap_NoninterestIncomeOtherOperatingIncome_291c0a8f-9d96-4fa1-ad28-308f18f34057" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BankOwnedLifeInsuranceIncome_809477f5-095c-4ee3-abe1-1e3df7c1d8e4" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BankOwnedLifeInsuranceIncome"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestIncome_6939bd84-fca8-4cd7-b64e-6de358c8bf19" xlink:to="loc_us-gaap_BankOwnedLifeInsuranceIncome_809477f5-095c-4ee3-abe1-1e3df7c1d8e4" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax_d638c579-2b07-4e28-bdf7-aeda4e73dcd5" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestIncome_6939bd84-fca8-4cd7-b64e-6de358c8bf19" xlink:to="loc_us-gaap_RevenueFromContractWithCustomerExcludingAssessedTax_d638c579-2b07-4e28-bdf7-aeda4e73dcd5" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeExpenseAfterProvisionForLoanLoss_196682e8-2e5f-43bd-8c0e-11e9cbdb058a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeExpenseAfterProvisionForLoanLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeExpenseNet_8169c84f-d71d-498e-87e0-66232dc37e6c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeExpenseNet"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeExpenseAfterProvisionForLoanLoss_196682e8-2e5f-43bd-8c0e-11e9cbdb058a" xlink:to="loc_us-gaap_InterestIncomeExpenseNet_8169c84f-d71d-498e-87e0-66232dc37e6c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_CreditLossExpenseReversal_c2d02afc-7e7c-40cc-8ebd-48a48a4a3635" xlink:href="bhrb-20250630.xsd#bhrb_CreditLossExpenseReversal"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeExpenseAfterProvisionForLoanLoss_196682e8-2e5f-43bd-8c0e-11e9cbdb058a" xlink:to="loc_bhrb_CreditLossExpenseReversal_c2d02afc-7e7c-40cc-8ebd-48a48a4a3635" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeOperating_df055425-5fe1-4674-aebf-f8d72ce6b95b" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeOperating"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_InterestAndFeeIncomeLoansAndLeasesTaxExempt_8d229fb5-a7e2-4fef-a6fe-a00f4538864f" xlink:href="bhrb-20250630.xsd#bhrb_InterestAndFeeIncomeLoansAndLeasesTaxExempt"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeOperating_df055425-5fe1-4674-aebf-f8d72ce6b95b" xlink:to="loc_bhrb_InterestAndFeeIncomeLoansAndLeasesTaxExempt_8d229fb5-a7e2-4fef-a6fe-a00f4538864f" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeOther_f4a6bc77-0374-4761-8131-63d3f0d98026" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeOther"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeOperating_df055425-5fe1-4674-aebf-f8d72ce6b95b" xlink:to="loc_us-gaap_InterestIncomeOther_f4a6bc77-0374-4761-8131-63d3f0d98026" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeSecuritiesTaxExempt_7927d757-5816-410b-aa1a-eb3fc659df70" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeSecuritiesTaxExempt"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeOperating_df055425-5fe1-4674-aebf-f8d72ce6b95b" xlink:to="loc_us-gaap_InterestIncomeSecuritiesTaxExempt_7927d757-5816-410b-aa1a-eb3fc659df70" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestAndFeeIncomeLoansAndLeases_27ce6269-ad1d-45ff-9523-ed4fde219471" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestAndFeeIncomeLoansAndLeases"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeOperating_df055425-5fe1-4674-aebf-f8d72ce6b95b" xlink:to="loc_us-gaap_InterestAndFeeIncomeLoansAndLeases_27ce6269-ad1d-45ff-9523-ed4fde219471" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeSecuritiesTaxable_89a42095-0e1d-4f78-ab9c-56fe5bc619d7" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeSecuritiesTaxable"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestIncomeOperating_df055425-5fe1-4674-aebf-f8d72ce6b95b" xlink:to="loc_us-gaap_InterestIncomeSecuritiesTaxable_89a42095-0e1d-4f78-ab9c-56fe5bc619d7" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoninterestExpense_7318ef48-902a-4d63-9a13-cb453deedffd" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoninterestExpense"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SalariesWagesAndOfficersCompensation_4190e23a-c57b-4351-ba77-4e02823ef4c7" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SalariesWagesAndOfficersCompensation"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestExpense_7318ef48-902a-4d63-9a13-cb453deedffd" xlink:to="loc_us-gaap_SalariesWagesAndOfficersCompensation_4190e23a-c57b-4351-ba77-4e02823ef4c7" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OccupancyNet_1191f86a-25c0-4d2c-a4a4-15ccca241b21" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OccupancyNet"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestExpense_7318ef48-902a-4d63-9a13-cb453deedffd" xlink:to="loc_us-gaap_OccupancyNet_1191f86a-25c0-4d2c-a4a4-15ccca241b21" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherNoninterestExpense_bd57a59f-d1ca-4dd1-9b23-1c79de8e16b1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherNoninterestExpense"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestExpense_7318ef48-902a-4d63-9a13-cb453deedffd" xlink:to="loc_us-gaap_OtherNoninterestExpense_bd57a59f-d1ca-4dd1-9b23-1c79de8e16b1" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquipmentExpense_71d73944-a404-4f02-9cd1-0507926409d6" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquipmentExpense"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestExpense_7318ef48-902a-4d63-9a13-cb453deedffd" xlink:to="loc_us-gaap_EquipmentExpense_71d73944-a404-4f02-9cd1-0507926409d6" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EmployeeBenefitsAndShareBasedCompensation_c28ab34d-2a43-45cf-8b46-d67f4efdb655" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EmployeeBenefitsAndShareBasedCompensation"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NoninterestExpense_7318ef48-902a-4d63-9a13-cb453deedffd" xlink:to="loc_us-gaap_EmployeeBenefitsAndShareBasedCompensation_c28ab34d-2a43-45cf-8b46-d67f4efdb655" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestExpenseOperating_96aa999a-a17b-4b69-b1ae-fdcda53e6a7f" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestExpenseOperating"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestExpenseSubordinatedNotesAndDebentures_66663520-f704-4a9f-a597-c5c200d87890" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestExpenseSubordinatedNotesAndDebentures"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestExpenseOperating_96aa999a-a17b-4b69-b1ae-fdcda53e6a7f" xlink:to="loc_us-gaap_InterestExpenseSubordinatedNotesAndDebentures_66663520-f704-4a9f-a597-c5c200d87890" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestExpenseShortTermBorrowings_1518eb38-dd48-4e92-af3e-9fe1f06bce05" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestExpenseShortTermBorrowings"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestExpenseOperating_96aa999a-a17b-4b69-b1ae-fdcda53e6a7f" xlink:to="loc_us-gaap_InterestExpenseShortTermBorrowings_1518eb38-dd48-4e92-af3e-9fe1f06bce05" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestExpenseDeposits_7deae47c-6068-4e42-87b4-227e297f2019" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestExpenseDeposits"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestExpenseOperating_96aa999a-a17b-4b69-b1ae-fdcda53e6a7f" xlink:to="loc_us-gaap_InterestExpenseDeposits_7deae47c-6068-4e42-87b4-227e297f2019" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestExpenseOther_581b88c6-c117-4c59-9ebc-a69ae4c0e212" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestExpenseOther"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_InterestExpenseOperating_96aa999a-a17b-4b69-b1ae-fdcda53e6a7f" xlink:to="loc_us-gaap_InterestExpenseOther_581b88c6-c117-4c59-9ebc-a69ae4c0e212" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLoss_c74e6354-1c1d-4068-9b3e-6468483561fe" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_277c8c20-993f-4d8b-af5f-8f86643e506e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetIncomeLoss_c74e6354-1c1d-4068-9b3e-6468483561fe" xlink:to="loc_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_277c8c20-993f-4d8b-af5f-8f86643e506e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeTaxExpenseBenefit_9146a9f0-a630-4ab0-a222-e992a4096995" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeTaxExpenseBenefit"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetIncomeLoss_c74e6354-1c1d-4068-9b3e-6468483561fe" xlink:to="loc_us-gaap_IncomeTaxExpenseBenefit_9146a9f0-a630-4ab0-a222-e992a4096995" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_a94b5c25-0bef-4ae3-a8da-3eea6427711f" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestIncomeExpenseAfterProvisionForLoanLoss_87f4546e-3ba0-400b-81a5-9b7c02f46068" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestIncomeExpenseAfterProvisionForLoanLoss"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_a94b5c25-0bef-4ae3-a8da-3eea6427711f" xlink:to="loc_us-gaap_InterestIncomeExpenseAfterProvisionForLoanLoss_87f4546e-3ba0-400b-81a5-9b7c02f46068" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoninterestIncome_c23e9aba-0b91-4db7-836c-4b4aa45e6f26" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoninterestIncome"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_a94b5c25-0bef-4ae3-a8da-3eea6427711f" xlink:to="loc_us-gaap_NoninterestIncome_c23e9aba-0b91-4db7-836c-4b4aa45e6f26" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoninterestExpense_9c0d6bd4-4e49-44e2-9b30-d19c7c2d8c78" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoninterestExpense"/>
    <link:calculationArc order="3" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_IncomeLossFromContinuingOperationsBeforeIncomeTaxesExtraordinaryItemsNoncontrollingInterest_a94b5c25-0bef-4ae3-a8da-3eea6427711f" xlink:to="loc_us-gaap_NoninterestExpense_9c0d6bd4-4e49-44e2-9b30-d19c7c2d8c78" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_CreditLossExpenseReversal_3ff43b73-de7d-442d-968f-a9923c17ed15" xlink:href="bhrb-20250630.xsd#bhrb_CreditLossExpenseReversal"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_FinancingReceivableExcludingAccruedInterestOffBalanceSheetCreditLossExpenseReversal_04ad0dfc-7cc8-4669-9323-0914608e47ad" xlink:href="bhrb-20250630.xsd#bhrb_FinancingReceivableExcludingAccruedInterestOffBalanceSheetCreditLossExpenseReversal"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_CreditLossExpenseReversal_3ff43b73-de7d-442d-968f-a9923c17ed15" xlink:to="loc_bhrb_FinancingReceivableExcludingAccruedInterestOffBalanceSheetCreditLossExpenseReversal_04ad0dfc-7cc8-4669-9323-0914608e47ad" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_FinancingReceivableExcludingAccruedInterestCreditLossExpenseReversalLoansAndDebtSecuritiesAvailableForSale_346c7748-3a09-4439-af4c-10fe22a21bb7" xlink:href="bhrb-20250630.xsd#bhrb_FinancingReceivableExcludingAccruedInterestCreditLossExpenseReversalLoansAndDebtSecuritiesAvailableForSale"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_CreditLossExpenseReversal_3ff43b73-de7d-442d-968f-a9923c17ed15" xlink:to="loc_bhrb_FinancingReceivableExcludingAccruedInterestCreditLossExpenseReversalLoansAndDebtSecuritiesAvailableForSale_346c7748-3a09-4439-af4c-10fe22a21bb7" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/ConsolidatedStatementsofComprehensiveIncomeLoss" xlink:type="simple" xlink:href="bhrb-20250630.xsd#ConsolidatedStatementsofComprehensiveIncomeLoss"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/ConsolidatedStatementsofComprehensiveIncomeLoss" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_737db6c3-0d34-4707-a3d0-42944e748bed" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeLossCashFlowHedgeGainLossReclassificationAfterTax_279a7186-1390-42a5-985d-ae4b9ba41652" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeLossCashFlowHedgeGainLossReclassificationAfterTax"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_737db6c3-0d34-4707-a3d0-42944e748bed" xlink:to="loc_us-gaap_OtherComprehensiveIncomeLossCashFlowHedgeGainLossReclassificationAfterTax_279a7186-1390-42a5-985d-ae4b9ba41652" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeLossDerivativeExcludedComponentIncreaseDecreaseAdjustmentsAfterTax_7f6a899e-46bc-4a87-b07d-f9402d58d837" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeLossDerivativeExcludedComponentIncreaseDecreaseAdjustmentsAfterTax"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_737db6c3-0d34-4707-a3d0-42944e748bed" xlink:to="loc_us-gaap_OtherComprehensiveIncomeLossDerivativeExcludedComponentIncreaseDecreaseAdjustmentsAfterTax_7f6a899e-46bc-4a87-b07d-f9402d58d837" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeLossReclassificationAdjustmentFromAOCIForSaleOfSecuritiesNetOfTax_09566571-2b2f-45d1-b57f-1a2c00d1593c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeLossReclassificationAdjustmentFromAOCIForSaleOfSecuritiesNetOfTax"/>
    <link:calculationArc order="3" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_737db6c3-0d34-4707-a3d0-42944e748bed" xlink:to="loc_us-gaap_OtherComprehensiveIncomeLossReclassificationAdjustmentFromAOCIForSaleOfSecuritiesNetOfTax_09566571-2b2f-45d1-b57f-1a2c00d1593c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeUnrealizedHoldingGainLossOnSecuritiesArisingDuringPeriodNetOfTax_4aaf1811-5d4f-4704-8f0c-95eebebb9620" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeUnrealizedHoldingGainLossOnSecuritiesArisingDuringPeriodNetOfTax"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_737db6c3-0d34-4707-a3d0-42944e748bed" xlink:to="loc_us-gaap_OtherComprehensiveIncomeUnrealizedHoldingGainLossOnSecuritiesArisingDuringPeriodNetOfTax_4aaf1811-5d4f-4704-8f0c-95eebebb9620" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeLossCashFlowHedgeGainLossBeforeReclassificationAfterTax_8f07a111-499d-4b4b-b0d7-22d49389604a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeLossCashFlowHedgeGainLossBeforeReclassificationAfterTax"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_737db6c3-0d34-4707-a3d0-42944e748bed" xlink:to="loc_us-gaap_OtherComprehensiveIncomeLossCashFlowHedgeGainLossBeforeReclassificationAfterTax_8f07a111-499d-4b4b-b0d7-22d49389604a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeDefinedBenefitPlansAdjustmentNetOfTaxPortionAttributableToParent_e5d592c2-13bd-45f8-985d-de283570018f" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeDefinedBenefitPlansAdjustmentNetOfTaxPortionAttributableToParent"/>
    <link:calculationArc order="6" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_737db6c3-0d34-4707-a3d0-42944e748bed" xlink:to="loc_us-gaap_OtherComprehensiveIncomeDefinedBenefitPlansAdjustmentNetOfTaxPortionAttributableToParent_e5d592c2-13bd-45f8-985d-de283570018f" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ComprehensiveIncomeNetOfTax_ef21c2de-9ad0-4f49-974c-928de466d5a3" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ComprehensiveIncomeNetOfTax"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_7e1b6bd1-7d34-4590-8569-978b112369de" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_ComprehensiveIncomeNetOfTax_ef21c2de-9ad0-4f49-974c-928de466d5a3" xlink:to="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_7e1b6bd1-7d34-4590-8569-978b112369de" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLoss_7d43abde-068d-487e-9c33-9a95e00b695e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLoss"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_ComprehensiveIncomeNetOfTax_ef21c2de-9ad0-4f49-974c-928de466d5a3" xlink:to="loc_us-gaap_NetIncomeLoss_7d43abde-068d-487e-9c33-9a95e00b695e" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/ConsolidatedStatementsofCashFlows" xlink:type="simple" xlink:href="bhrb-20250630.xsd#ConsolidatedStatementsofCashFlows"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/ConsolidatedStatementsofCashFlows" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInFinancingActivities"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeasePrincipalPayments_d0c53706-8775-42e3-9edd-1e40b95bc443" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeasePrincipalPayments"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:to="loc_us-gaap_FinanceLeasePrincipalPayments_d0c53706-8775-42e3-9edd-1e40b95bc443" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsOfDividends_eefbd823-8a74-4a9c-9862-45ee28dc0e4b" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsOfDividends"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:to="loc_us-gaap_PaymentsOfDividends_eefbd823-8a74-4a9c-9862-45ee28dc0e4b" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromIssuanceOfCommonStock_f077b139-56c9-45ec-9554-fca0abad5be9" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromIssuanceOfCommonStock"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:to="loc_us-gaap_ProceedsFromIssuanceOfCommonStock_f077b139-56c9-45ec-9554-fca0abad5be9" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromStockPlans_36483850-719f-4853-a324-7014dc0aee04" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromStockPlans"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:to="loc_us-gaap_ProceedsFromStockPlans_36483850-719f-4853-a324-7014dc0aee04" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromRepaymentsOfOtherDebt_4dcf61a6-2dca-4278-81c4-1f93d622c1a0" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromRepaymentsOfOtherDebt"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:to="loc_us-gaap_ProceedsFromRepaymentsOfOtherDebt_4dcf61a6-2dca-4278-81c4-1f93d622c1a0" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromSaleOfTreasuryStock_c9a11e77-8253-4900-acce-f856c7f5dbad" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromSaleOfTreasuryStock"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:to="loc_us-gaap_ProceedsFromSaleOfTreasuryStock_c9a11e77-8253-4900-acce-f856c7f5dbad" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetChangeNoninterestBearingDepositsDomestic_20b6f3f7-e538-4f71-ab22-8b889f85b7cb" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetChangeNoninterestBearingDepositsDomestic"/>
    <link:calculationArc order="7" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:to="loc_us-gaap_NetChangeNoninterestBearingDepositsDomestic_20b6f3f7-e538-4f71-ab22-8b889f85b7cb" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetChangeInterestBearingDepositsDomestic_2c65755f-8475-42bd-b84d-787658e8f92d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetChangeInterestBearingDepositsDomestic"/>
    <link:calculationArc order="8" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_63f10527-4c94-4a1f-8035-b4ca1cc7cbf9" xlink:to="loc_us-gaap_NetChangeInterestBearingDepositsDomestic_2c65755f-8475-42bd-b84d-787658e8f92d" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_f5bf4959-330b-40cb-a902-9e90a744c154" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_98de8003-a02e-4277-a6d7-4c7e52e87d43" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInOperatingActivities"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_f5bf4959-330b-40cb-a902-9e90a744c154" xlink:to="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_98de8003-a02e-4277-a6d7-4c7e52e87d43" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_a7c87377-7dea-454b-8b60-990c9fdb8b51" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInInvestingActivities"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_f5bf4959-330b-40cb-a902-9e90a744c154" xlink:to="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_a7c87377-7dea-454b-8b60-990c9fdb8b51" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_8b3f7d33-8859-4562-b969-2b9bacd530cc" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInFinancingActivities"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsPeriodIncreaseDecreaseIncludingExchangeRateEffect_f5bf4959-330b-40cb-a902-9e90a744c154" xlink:to="loc_us-gaap_NetCashProvidedByUsedInFinancingActivities_8b3f7d33-8859-4562-b969-2b9bacd530cc" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInInvestingActivities"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsToAcquireRestrictedInvestments_d59c16f3-7727-4c50-b423-c340af24153e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsToAcquireRestrictedInvestments"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_PaymentsToAcquireRestrictedInvestments_d59c16f3-7727-4c50-b423-c340af24153e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_IncreaseDecreaseInCustomerLoans_2e8abc44-c7fa-41de-9581-a44b2a815f50" xlink:href="bhrb-20250630.xsd#bhrb_IncreaseDecreaseInCustomerLoans"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_bhrb_IncreaseDecreaseInCustomerLoans_2e8abc44-c7fa-41de-9581-a44b2a815f50" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromMaturitiesPrepaymentsAndCallsOfAvailableForSaleSecurities_6890f186-60e8-4aa7-8fea-38668efe9fb9" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromMaturitiesPrepaymentsAndCallsOfAvailableForSaleSecurities"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_ProceedsFromMaturitiesPrepaymentsAndCallsOfAvailableForSaleSecurities_6890f186-60e8-4aa7-8fea-38668efe9fb9" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_9ca86cf9-f91f-47c4-9b2f-7787232103c0" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsToAcquirePropertyPlantAndEquipment"/>
    <link:calculationArc order="4" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_PaymentsToAcquirePropertyPlantAndEquipment_9ca86cf9-f91f-47c4-9b2f-7787232103c0" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromSaleOfRestrictedInvestments_66c16e12-e8d1-42eb-96e3-2ecd9be85832" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromSaleOfRestrictedInvestments"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_ProceedsFromSaleOfRestrictedInvestments_66c16e12-e8d1-42eb-96e3-2ecd9be85832" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromSaleOfAvailableForSaleSecuritiesDebt_6c713b5b-c3d4-4098-a215-e00d245da5fe" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromSaleOfAvailableForSaleSecuritiesDebt"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_ProceedsFromSaleOfAvailableForSaleSecuritiesDebt_6c713b5b-c3d4-4098-a215-e00d245da5fe" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsToAcquireAvailableForSaleSecuritiesDebt_b8d2ad94-14b2-4180-aae1-84035fc64eb9" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsToAcquireAvailableForSaleSecuritiesDebt"/>
    <link:calculationArc order="7" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_PaymentsToAcquireAvailableForSaleSecuritiesDebt_b8d2ad94-14b2-4180-aae1-84035fc64eb9" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsForProceedsFromLifeInsurancePolicies_2722d843-5302-4ba1-8e0c-93a3ac57c621" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsForProceedsFromLifeInsurancePolicies"/>
    <link:calculationArc order="8" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_PaymentsForProceedsFromLifeInsurancePolicies_2722d843-5302-4ba1-8e0c-93a3ac57c621" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsToAcquireBusinessesNetOfCashAcquired_880e09a3-8fcb-4729-88c1-6f9c2b0cd433" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsToAcquireBusinessesNetOfCashAcquired"/>
    <link:calculationArc order="9" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_PaymentsToAcquireBusinessesNetOfCashAcquired_880e09a3-8fcb-4729-88c1-6f9c2b0cd433" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromSaleOfOtherRealEstate_2d7a9c3a-f46e-4d12-be5f-d9ea9620b164" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromSaleOfOtherRealEstate"/>
    <link:calculationArc order="10" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInInvestingActivities_83f9004c-9a71-4fa5-a62a-452fc5c148de" xlink:to="loc_us-gaap_ProceedsFromSaleOfOtherRealEstate_2d7a9c3a-f46e-4d12-be5f-d9ea9620b164" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInOperatingActivities"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_IncreaseDecreaseInInterestPayableAndOtherLiabilities_6014ed05-a766-4c00-9a86-e76338296230" xlink:href="bhrb-20250630.xsd#bhrb_IncreaseDecreaseInInterestPayableAndOtherLiabilities"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_IncreaseDecreaseInInterestPayableAndOtherLiabilities_6014ed05-a766-4c00-9a86-e76338296230" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_GainsLossesOnSalesOfOtherRealEstate_8a193085-4da2-40fb-b6ca-c4ed78d00c89" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_GainsLossesOnSalesOfOtherRealEstate"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_GainsLossesOnSalesOfOtherRealEstate_8a193085-4da2-40fb-b6ca-c4ed78d00c89" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromSaleOfLoansHeldForSale_2b236bbc-9911-42cc-8539-03b29882d64a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromSaleOfLoansHeldForSale"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_ProceedsFromSaleOfLoansHeldForSale_2b236bbc-9911-42cc-8539-03b29882d64a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_AccretionIncomeFromAcquiredLoans_9406c337-e77a-4981-ae2b-280c0ebe6ff9" xlink:href="bhrb-20250630.xsd#bhrb_AccretionIncomeFromAcquiredLoans"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_AccretionIncomeFromAcquiredLoans_9406c337-e77a-4981-ae2b-280c0ebe6ff9" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_GainLossOnSalesOfLoansNet_bb3f9794-953d-4827-a60e-90f10b7098c6" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_GainLossOnSalesOfLoansNet"/>
    <link:calculationArc order="5" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_GainLossOnSalesOfLoansNet_bb3f9794-953d-4827-a60e-90f10b7098c6" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInOtherOperatingAssets_56efef33-4303-491e-a9e3-7434b80f634d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInOtherOperatingAssets"/>
    <link:calculationArc order="6" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_IncreaseDecreaseInOtherOperatingAssets_56efef33-4303-491e-a9e3-7434b80f634d" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DeferredIncomeTaxExpenseBenefit_1df19c7a-4771-40e1-8f5a-d11d58f5be0e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DeferredIncomeTaxExpenseBenefit"/>
    <link:calculationArc order="7" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_DeferredIncomeTaxExpenseBenefit_1df19c7a-4771-40e1-8f5a-d11d58f5be0e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_AmortizationOfDiscountsAndPremiumsInvestments_22716749-9129-46ec-8e37-b952e0f45be1" xlink:href="bhrb-20250630.xsd#bhrb_AmortizationOfDiscountsAndPremiumsInvestments"/>
    <link:calculationArc order="8" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_AmortizationOfDiscountsAndPremiumsInvestments_22716749-9129-46ec-8e37-b952e0f45be1" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_AmortizationOnAssumedLiabilities_d2de456c-9f39-4b28-9f83-cdbc3701e527" xlink:href="bhrb-20250630.xsd#bhrb_AmortizationOnAssumedLiabilities"/>
    <link:calculationArc order="9" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_AmortizationOnAssumedLiabilities_d2de456c-9f39-4b28-9f83-cdbc3701e527" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DepreciationDepletionAndAmortization_9feb95ca-7b1a-44ad-942b-1e0bf730e156" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DepreciationDepletionAndAmortization"/>
    <link:calculationArc order="10" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_DepreciationDepletionAndAmortization_9feb95ca-7b1a-44ad-942b-1e0bf730e156" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_GainLossOnLoansHeldForSaleFairValueAdjustment_67b6314a-4dd3-4e0c-a777-a8977da4c9ac" xlink:href="bhrb-20250630.xsd#bhrb_GainLossOnLoansHeldForSaleFairValueAdjustment"/>
    <link:calculationArc order="11" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_GainLossOnLoansHeldForSaleFairValueAdjustment_67b6314a-4dd3-4e0c-a777-a8977da4c9ac" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtAndEquitySecuritiesGainLoss_db85be66-2995-4cd0-8650-ca982bd0e1f3" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtAndEquitySecuritiesGainLoss"/>
    <link:calculationArc order="12" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_DebtAndEquitySecuritiesGainLoss_db85be66-2995-4cd0-8650-ca982bd0e1f3" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BankOwnedLifeInsuranceIncome_817750b5-2d7d-4ec4-b066-6441edfb5554" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BankOwnedLifeInsuranceIncome"/>
    <link:calculationArc order="13" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_BankOwnedLifeInsuranceIncome_817750b5-2d7d-4ec4-b066-6441edfb5554" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_AmortizationOfHousingTaxCredits_06edf17f-6c0c-48e1-aa9c-e5ac6c00c2c9" xlink:href="bhrb-20250630.xsd#bhrb_AmortizationOfHousingTaxCredits"/>
    <link:calculationArc order="14" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_AmortizationOfHousingTaxCredits_06edf17f-6c0c-48e1-aa9c-e5ac6c00c2c9" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_AccretionOfDiscountsAndPremiumsInvestments_c436dd7d-afa8-41fc-9df4-98627605896e" xlink:href="bhrb-20250630.xsd#bhrb_AccretionOfDiscountsAndPremiumsInvestments"/>
    <link:calculationArc order="15" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_AccretionOfDiscountsAndPremiumsInvestments_c436dd7d-afa8-41fc-9df4-98627605896e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInAccruedInterestReceivableNet_976aef28-a73b-4c3e-a55f-e30a0228ebc4" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInAccruedInterestReceivableNet"/>
    <link:calculationArc order="16" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_IncreaseDecreaseInAccruedInterestReceivableNet_976aef28-a73b-4c3e-a55f-e30a0228ebc4" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensation_87958a50-6e16-4cfb-904e-4459ef46fc88" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensation"/>
    <link:calculationArc order="17" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_ShareBasedCompensation_87958a50-6e16-4cfb-904e-4459ef46fc88" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsForOriginationAndPurchasesOfLoansHeldForSale_a2a46088-0a77-43b8-9ea0-41438c003d87" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsForOriginationAndPurchasesOfLoansHeldForSale"/>
    <link:calculationArc order="18" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_PaymentsForOriginationAndPurchasesOfLoansHeldForSale_a2a46088-0a77-43b8-9ea0-41438c003d87" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_CreditLossExpenseReversal_8eb966b8-6062-4f1e-b574-caf543dd249e" xlink:href="bhrb-20250630.xsd#bhrb_CreditLossExpenseReversal"/>
    <link:calculationArc order="19" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_CreditLossExpenseReversal_8eb966b8-6062-4f1e-b574-caf543dd249e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_GainLossOnSaleOfPropertyPlantEquipment_ef87c354-50f9-4de0-9530-88b481ceabc4" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_GainLossOnSaleOfPropertyPlantEquipment"/>
    <link:calculationArc order="20" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_GainLossOnSaleOfPropertyPlantEquipment_ef87c354-50f9-4de0-9530-88b481ceabc4" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInOperatingLeaseLiability_c27e5110-8cc6-4900-9115-8d4a20a620fe" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInOperatingLeaseLiability"/>
    <link:calculationArc order="21" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_IncreaseDecreaseInOperatingLeaseLiability_c27e5110-8cc6-4900-9115-8d4a20a620fe" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLoss_9ee53ce3-e80d-4d60-a822-773a59edcddd" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLoss"/>
    <link:calculationArc order="22" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_us-gaap_NetIncomeLoss_9ee53ce3-e80d-4d60-a822-773a59edcddd" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_AmortizationOfIntangibleAssetsAndRightOfUseAssets_bb75dc21-8d70-46e0-af9a-bc4a9cdaa3c6" xlink:href="bhrb-20250630.xsd#bhrb_AmortizationOfIntangibleAssetsAndRightOfUseAssets"/>
    <link:calculationArc order="23" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivities_0082129c-a7aa-4bf8-825b-f25b8e3f4a2f" xlink:to="loc_bhrb_AmortizationOfIntangibleAssetsAndRightOfUseAssets_bb75dc21-8d70-46e0-af9a-bc4a9cdaa3c6" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/SecuritiesScheduleofDebtSecuritiesAvailableforSaleDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#SecuritiesScheduleofDebtSecuritiesAvailableforSaleDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/SecuritiesScheduleofDebtSecuritiesAvailableforSaleDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_566926fb-7fca-42ac-825f-8bff0ccaeeb3" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleDebtSecuritiesAccumulatedGrossUnrealizedGainBeforeTax_4fd8cc78-7747-4bd0-ac69-3f7ef8be5a4d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleDebtSecuritiesAccumulatedGrossUnrealizedGainBeforeTax"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_566926fb-7fca-42ac-825f-8bff0ccaeeb3" xlink:to="loc_us-gaap_AvailableForSaleDebtSecuritiesAccumulatedGrossUnrealizedGainBeforeTax_4fd8cc78-7747-4bd0-ac69-3f7ef8be5a4d" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleDebtSecuritiesAccumulatedGrossUnrealizedLossBeforeTax_1dd731f5-eabb-4afb-9b6f-f00b69bbb872" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleDebtSecuritiesAccumulatedGrossUnrealizedLossBeforeTax"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_566926fb-7fca-42ac-825f-8bff0ccaeeb3" xlink:to="loc_us-gaap_AvailableForSaleDebtSecuritiesAccumulatedGrossUnrealizedLossBeforeTax_1dd731f5-eabb-4afb-9b6f-f00b69bbb872" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_b6fc3803-d12a-4978-b4ce-90ab488694a8" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_566926fb-7fca-42ac-825f-8bff0ccaeeb3" xlink:to="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_b6fc3803-d12a-4978-b4ce-90ab488694a8" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/SecuritiesScheduleofExpectedMaturityforDebtSecuritiesAvailableforSaleDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#SecuritiesScheduleofExpectedMaturityforDebtSecuritiesAvailableforSaleDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/SecuritiesScheduleofExpectedMaturityforDebtSecuritiesAvailableforSaleDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_570063bc-a9d7-4b48-8c1a-31ac37041eb0" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterTenYearsFairValue_4fabbc89-2296-41f1-8d32-e59d470c635c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterTenYearsFairValue"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_570063bc-a9d7-4b48-8c1a-31ac37041eb0" xlink:to="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterTenYearsFairValue_4fabbc89-2296-41f1-8d32-e59d470c635c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue_3710669d-4a84-48e7-8dde-9cf4fc04ca7d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_570063bc-a9d7-4b48-8c1a-31ac37041eb0" xlink:to="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue_3710669d-4a84-48e7-8dde-9cf4fc04ca7d" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterOneThroughFiveYearsFairValue_dbc52dd4-1138-422a-894d-398004458da2" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterOneThroughFiveYearsFairValue"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_570063bc-a9d7-4b48-8c1a-31ac37041eb0" xlink:to="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterOneThroughFiveYearsFairValue_dbc52dd4-1138-422a-894d-398004458da2" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearFairValue_454719c9-b956-42b7-948b-673dd1bc94de" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearFairValue"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleExcludingAccruedInterest_570063bc-a9d7-4b48-8c1a-31ac37041eb0" xlink:to="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearFairValue_454719c9-b956-42b7-948b-673dd1bc94de" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_16002ffe-8d0e-49c3-9aa1-bfc505ff9b94" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterTenYearsAmortizedCost_1a3ec186-620b-40ee-9462-697570b631ad" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterTenYearsAmortizedCost"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_16002ffe-8d0e-49c3-9aa1-bfc505ff9b94" xlink:to="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterTenYearsAmortizedCost_1a3ec186-620b-40ee-9462-697570b631ad" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsAmortizedCost_9bcf8e9c-7394-4b7a-9e11-7dd2a8706d89" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsAmortizedCost"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_16002ffe-8d0e-49c3-9aa1-bfc505ff9b94" xlink:to="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsAmortizedCost_9bcf8e9c-7394-4b7a-9e11-7dd2a8706d89" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterOneThroughFiveYearsAmortizedCost_4f043329-aef3-4483-8cd6-4d0945c3f78e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterOneThroughFiveYearsAmortizedCost"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_16002ffe-8d0e-49c3-9aa1-bfc505ff9b94" xlink:to="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterOneThroughFiveYearsAmortizedCost_4f043329-aef3-4483-8cd6-4d0945c3f78e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearAmortizedCost_b197190b-d046-4311-b1f5-0aa8e9edbf70" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearAmortizedCost"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleAmortizedCostExcludingAccruedInterestAfterAllowanceForCreditLoss_16002ffe-8d0e-49c3-9aa1-bfc505ff9b94" xlink:to="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearAmortizedCost_b197190b-d046-4311-b1f5-0aa8e9edbf70" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/SecuritiesScheduleofDebtSecuritiesAvailableforSaleUnrealizedLossPositionFairValueDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#SecuritiesScheduleofDebtSecuritiesAvailableforSaleUnrealizedLossPositionFairValueDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/SecuritiesScheduleofDebtSecuritiesAvailableforSaleUnrealizedLossPositionFairValueDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleUnrealizedLossPositionAccumulatedLoss_9e53d583-7894-435a-a676-53b09b393a97" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleUnrealizedLossPositionAccumulatedLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleContinuousUnrealizedLossPosition12MonthsOrLongerAccumulatedLoss_6549a73e-58e1-4236-80ed-9488ab2eca8c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleContinuousUnrealizedLossPosition12MonthsOrLongerAccumulatedLoss"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleUnrealizedLossPositionAccumulatedLoss_9e53d583-7894-435a-a676-53b09b393a97" xlink:to="loc_us-gaap_DebtSecuritiesAvailableForSaleContinuousUnrealizedLossPosition12MonthsOrLongerAccumulatedLoss_6549a73e-58e1-4236-80ed-9488ab2eca8c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleContinuousUnrealizedLossPositionLessThan12MonthsAccumulatedLoss_1c2113a9-877b-4053-b507-bebd24627f6b" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleContinuousUnrealizedLossPositionLessThan12MonthsAccumulatedLoss"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleUnrealizedLossPositionAccumulatedLoss_9e53d583-7894-435a-a676-53b09b393a97" xlink:to="loc_us-gaap_DebtSecuritiesAvailableForSaleContinuousUnrealizedLossPositionLessThan12MonthsAccumulatedLoss_1c2113a9-877b-4053-b507-bebd24627f6b" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/AllowanceforCreditLossesScheduleofLoansbyCreditQualityIndicatorsDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#AllowanceforCreditLossesScheduleofLoansbyCreditQualityIndicatorsDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/AllowanceforCreditLossesScheduleofLoansbyCreditQualityIndicatorsDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff_39897e57-3134-449e-a613-3ca32d88ef05" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearThreeOriginatedTwoYearsBeforeCurrentFiscalYearWriteoff_680a7551-37a4-4eb8-9a3e-e2f817431af0" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearThreeOriginatedTwoYearsBeforeCurrentFiscalYearWriteoff"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff_39897e57-3134-449e-a613-3ca32d88ef05" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearThreeOriginatedTwoYearsBeforeCurrentFiscalYearWriteoff_680a7551-37a4-4eb8-9a3e-e2f817431af0" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearOneOriginatedCurrentFiscalYearWriteoff_9a7fede1-9570-4bb2-a993-3ee669494e9f" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearOneOriginatedCurrentFiscalYearWriteoff"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff_39897e57-3134-449e-a613-3ca32d88ef05" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearOneOriginatedCurrentFiscalYearWriteoff_9a7fede1-9570-4bb2-a993-3ee669494e9f" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestOriginatedMoreThanFiveYearsBeforeCurrentFiscalYearWriteoff_8bebbbe8-a46a-4073-be50-d3303c577ee5" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestOriginatedMoreThanFiveYearsBeforeCurrentFiscalYearWriteoff"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff_39897e57-3134-449e-a613-3ca32d88ef05" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestOriginatedMoreThanFiveYearsBeforeCurrentFiscalYearWriteoff_8bebbbe8-a46a-4073-be50-d3303c577ee5" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearFiveOriginatedFourYearsBeforeCurrentFiscalYearWriteoff_acb350c8-41b5-4b0c-aff9-82b00aeba387" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearFiveOriginatedFourYearsBeforeCurrentFiscalYearWriteoff"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff_39897e57-3134-449e-a613-3ca32d88ef05" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearFiveOriginatedFourYearsBeforeCurrentFiscalYearWriteoff_acb350c8-41b5-4b0c-aff9-82b00aeba387" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestRevolvingWriteoff_ba90e3a4-3121-401f-b34a-2a4603f24168" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestRevolvingWriteoff"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff_39897e57-3134-449e-a613-3ca32d88ef05" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestRevolvingWriteoff_ba90e3a4-3121-401f-b34a-2a4603f24168" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearFourOriginatedThreeYearsBeforeCurrentFiscalYearWriteoff_c73872cb-aa69-4eb0-ae8c-8b4c49ef6fc6" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearFourOriginatedThreeYearsBeforeCurrentFiscalYearWriteoff"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff_39897e57-3134-449e-a613-3ca32d88ef05" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearFourOriginatedThreeYearsBeforeCurrentFiscalYearWriteoff_c73872cb-aa69-4eb0-ae8c-8b4c49ef6fc6" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearTwoOriginatedFiscalYearBeforeCurrentFiscalYearWriteoff_ef79a6d5-f44f-49a5-a4d7-404fde067d97" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearTwoOriginatedFiscalYearBeforeCurrentFiscalYearWriteoff"/>
    <link:calculationArc order="7" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestAllowanceForCreditLossWriteoff_39897e57-3134-449e-a613-3ca32d88ef05" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearTwoOriginatedFiscalYearBeforeCurrentFiscalYearWriteoff_ef79a6d5-f44f-49a5-a4d7-404fde067d97" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_fbe45ef2-5d70-4a62-9acf-a1af1601587c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearThreeOriginatedTwoYearsBeforeCurrentFiscalYear_b7fbd05c-3db4-45d0-90b5-a5da7dc6bc61" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearThreeOriginatedTwoYearsBeforeCurrentFiscalYear"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_fbe45ef2-5d70-4a62-9acf-a1af1601587c" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearThreeOriginatedTwoYearsBeforeCurrentFiscalYear_b7fbd05c-3db4-45d0-90b5-a5da7dc6bc61" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestRevolving_1c27576c-ab81-43c7-a3ed-8a23ace1fd2f" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestRevolving"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_fbe45ef2-5d70-4a62-9acf-a1af1601587c" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestRevolving_1c27576c-ab81-43c7-a3ed-8a23ace1fd2f" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearOneOriginatedCurrentFiscalYear_3ee40ebc-f922-4a03-9e4d-d9f0fff0480a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearOneOriginatedCurrentFiscalYear"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_fbe45ef2-5d70-4a62-9acf-a1af1601587c" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearOneOriginatedCurrentFiscalYear_3ee40ebc-f922-4a03-9e4d-d9f0fff0480a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearFourOriginatedThreeYearsBeforeCurrentFiscalYear_6396111b-72ae-4dad-bc7c-9177632aa8a2" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearFourOriginatedThreeYearsBeforeCurrentFiscalYear"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_fbe45ef2-5d70-4a62-9acf-a1af1601587c" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearFourOriginatedThreeYearsBeforeCurrentFiscalYear_6396111b-72ae-4dad-bc7c-9177632aa8a2" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestOriginatedMoreThanFiveYearsBeforeCurrentFiscalYear_046c74ec-3506-4715-bf22-4bf7d992367a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestOriginatedMoreThanFiveYearsBeforeCurrentFiscalYear"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_fbe45ef2-5d70-4a62-9acf-a1af1601587c" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestOriginatedMoreThanFiveYearsBeforeCurrentFiscalYear_046c74ec-3506-4715-bf22-4bf7d992367a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearFiveOriginatedFourYearsBeforeCurrentFiscalYear_1042d015-cf06-4385-9f8d-a3ed0fb769ab" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearFiveOriginatedFourYearsBeforeCurrentFiscalYear"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_fbe45ef2-5d70-4a62-9acf-a1af1601587c" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearFiveOriginatedFourYearsBeforeCurrentFiscalYear_1042d015-cf06-4385-9f8d-a3ed0fb769ab" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearTwoOriginatedFiscalYearBeforeCurrentFiscalYear_e5dba14f-0231-4c79-abca-02a266305659" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableExcludingAccruedInterestYearTwoOriginatedFiscalYearBeforeCurrentFiscalYear"/>
    <link:calculationArc order="7" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivableExcludingAccruedInterestBeforeAllowanceForCreditLoss_fbe45ef2-5d70-4a62-9acf-a1af1601587c" xlink:to="loc_us-gaap_FinancingReceivableExcludingAccruedInterestYearTwoOriginatedFiscalYearBeforeCurrentFiscalYear_e5dba14f-0231-4c79-abca-02a266305659" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/AllowanceforCreditLossesScheduleofPurchasedCreditDeterioratedLoansDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#AllowanceforCreditLossesScheduleofPurchasedCreditDeterioratedLoansDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/AllowanceforCreditLossesScheduleofPurchasedCreditDeterioratedLoansDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAmountAtPurchasePrice_5524c1b3-c812-4803-8cc2-cf085aaceeef" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAmountAtPurchasePrice"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationDiscountPremium_dcf04faf-1774-43f8-b453-070a9fc7bee4" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationDiscountPremium"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAmountAtPurchasePrice_5524c1b3-c812-4803-8cc2-cf085aaceeef" xlink:to="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationDiscountPremium_dcf04faf-1774-43f8-b453-070a9fc7bee4" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAllowanceForCreditLossAtAcquisitionDate_030de392-3614-404c-b7fd-44bd28572f01" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAllowanceForCreditLossAtAcquisitionDate"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAmountAtPurchasePrice_5524c1b3-c812-4803-8cc2-cf085aaceeef" xlink:to="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAllowanceForCreditLossAtAcquisitionDate_030de392-3614-404c-b7fd-44bd28572f01" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAmountAtParValue_38808910-a803-4fa7-8360-749505d0bce9" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAmountAtParValue"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAmountAtPurchasePrice_5524c1b3-c812-4803-8cc2-cf085aaceeef" xlink:to="loc_us-gaap_FinancingReceivablePurchasedWithCreditDeteriorationAmountAtParValue_38808910-a803-4fa7-8360-749505d0bce9" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/DepositsScheduleofTimeDepositMaturitiesDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#DepositsScheduleofTimeDepositMaturitiesDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/DepositsScheduleofTimeDepositMaturitiesDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TimeDeposits_e8dbf278-daee-411f-86dc-f0445cd2e5fc" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TimeDeposits"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TimeDepositMaturitiesYearThree_6f0dc538-efd4-4b11-8e2f-009052af7d3e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TimeDepositMaturitiesYearThree"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_TimeDeposits_e8dbf278-daee-411f-86dc-f0445cd2e5fc" xlink:to="loc_us-gaap_TimeDepositMaturitiesYearThree_6f0dc538-efd4-4b11-8e2f-009052af7d3e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TimeDepositMaturitiesYearFour_21ce6273-8af4-4994-9a3b-4a4b74feffd3" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TimeDepositMaturitiesYearFour"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_TimeDeposits_e8dbf278-daee-411f-86dc-f0445cd2e5fc" xlink:to="loc_us-gaap_TimeDepositMaturitiesYearFour_21ce6273-8af4-4994-9a3b-4a4b74feffd3" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TimeDepositMaturitiesYearOne_5b5b018f-18e1-435f-9df2-0d99df909e23" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TimeDepositMaturitiesYearOne"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_TimeDeposits_e8dbf278-daee-411f-86dc-f0445cd2e5fc" xlink:to="loc_us-gaap_TimeDepositMaturitiesYearOne_5b5b018f-18e1-435f-9df2-0d99df909e23" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TimeDepositMaturitiesRemainderOfFiscalYear_25f836df-ce10-414d-ab65-b6e321108354" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TimeDepositMaturitiesRemainderOfFiscalYear"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_TimeDeposits_e8dbf278-daee-411f-86dc-f0445cd2e5fc" xlink:to="loc_us-gaap_TimeDepositMaturitiesRemainderOfFiscalYear_25f836df-ce10-414d-ab65-b6e321108354" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_TimeDepositMaturitiesAfterYearFour_2a1eac20-a4f3-4c45-95dd-7e546995db20" xlink:href="bhrb-20250630.xsd#bhrb_TimeDepositMaturitiesAfterYearFour"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_TimeDeposits_e8dbf278-daee-411f-86dc-f0445cd2e5fc" xlink:to="loc_bhrb_TimeDepositMaturitiesAfterYearFour_2a1eac20-a4f3-4c45-95dd-7e546995db20" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TimeDepositMaturitiesYearTwo_53854f50-112d-44b3-89ba-e09968001561" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TimeDepositMaturitiesYearTwo"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_TimeDeposits_e8dbf278-daee-411f-86dc-f0445cd2e5fc" xlink:to="loc_us-gaap_TimeDepositMaturitiesYearTwo_53854f50-112d-44b3-89ba-e09968001561" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/BorrowedFundsScheduleofContractualMaturitiesDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#BorrowedFundsScheduleofContractualMaturitiesDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/BorrowedFundsScheduleofContractualMaturitiesDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShortTermBorrowings_9513a5bf-9c48-4e82-b430-69fdd039b198" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShortTermBorrowings"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_ShortTermDebtMaturityRemainderOfFiscalYear_4a6e1854-6328-43be-afc3-a606e14ace95" xlink:href="bhrb-20250630.xsd#bhrb_ShortTermDebtMaturityRemainderOfFiscalYear"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_ShortTermBorrowings_9513a5bf-9c48-4e82-b430-69fdd039b198" xlink:to="loc_bhrb_ShortTermDebtMaturityRemainderOfFiscalYear_4a6e1854-6328-43be-afc3-a606e14ace95" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/BorrowedFundsScheduleofRemainingMaturitiesofSubordinatedDebenturesDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#BorrowedFundsScheduleofRemainingMaturitiesofSubordinatedDebenturesDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/BorrowedFundsScheduleofRemainingMaturitiesofSubordinatedDebenturesDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentCarryingAmount_a05cf3a8-e866-409b-bb5f-4a5fc109de15" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentCarryingAmount"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearThree_ced2caba-cf89-4899-b812-d7932fd1d4bf" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearThree"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtInstrumentCarryingAmount_a05cf3a8-e866-409b-bb5f-4a5fc109de15" xlink:to="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearThree_ced2caba-cf89-4899-b812-d7932fd1d4bf" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearTwo_3ca2e203-dda6-4587-b3c1-cfe40cfdd16c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearTwo"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtInstrumentCarryingAmount_a05cf3a8-e866-409b-bb5f-4a5fc109de15" xlink:to="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearTwo_3ca2e203-dda6-4587-b3c1-cfe40cfdd16c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_LongTermDebtMaturityAfterYearFour_57df89e9-ca20-48ab-8c37-1d746b7fdad7" xlink:href="bhrb-20250630.xsd#bhrb_LongTermDebtMaturityAfterYearFour"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtInstrumentCarryingAmount_a05cf3a8-e866-409b-bb5f-4a5fc109de15" xlink:to="loc_bhrb_LongTermDebtMaturityAfterYearFour_57df89e9-ca20-48ab-8c37-1d746b7fdad7" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearFour_95286c72-e96e-4e86-882c-d8017fe679a2" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearFour"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtInstrumentCarryingAmount_a05cf3a8-e866-409b-bb5f-4a5fc109de15" xlink:to="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInYearFour_95286c72-e96e-4e86-882c-d8017fe679a2" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalRemainderOfFiscalYear_f2f8e9e7-9170-446f-8661-97b2a2159d4d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalRemainderOfFiscalYear"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtInstrumentCarryingAmount_a05cf3a8-e866-409b-bb5f-4a5fc109de15" xlink:to="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalRemainderOfFiscalYear_f2f8e9e7-9170-446f-8661-97b2a2159d4d" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInNextTwelveMonths_9f34c752-0c11-4452-9afa-3aa8a98fa237" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInNextTwelveMonths"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_DebtInstrumentCarryingAmount_a05cf3a8-e866-409b-bb5f-4a5fc109de15" xlink:to="loc_us-gaap_LongTermDebtMaturitiesRepaymentsOfPrincipalInNextTwelveMonths_9f34c752-0c11-4452-9afa-3aa8a98fa237" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofLeaseIncomeIncludedinNonInterestExpenseDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#LeasedPropertyScheduleofLeaseIncomeIncludedinNonInterestExpenseDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofLeaseIncomeIncludedinNonInterestExpenseDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LeaseIncome_5a05f036-e5ab-40ff-b556-c7fe48ed35bb" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LeaseIncome"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseLeaseIncome_7ed8bdbf-16a1-4e18-a453-a04888cf2512" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseLeaseIncome"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LeaseIncome_5a05f036-e5ab-40ff-b556-c7fe48ed35bb" xlink:to="loc_us-gaap_OperatingLeaseLeaseIncome_7ed8bdbf-16a1-4e18-a453-a04888cf2512" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofRemainingMaturitiesofOperatingLeaseReceivablesDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#LeasedPropertyScheduleofRemainingMaturitiesofOperatingLeaseReceivablesDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofRemainingMaturitiesofOperatingLeaseReceivablesDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceived_1b1f59e1-cfee-488c-bbb2-c9c4dad4ced2" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LessorOperatingLeasePaymentsToBeReceived"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedThreeYears_b2da1b65-c218-4a77-986c-c289ff107013" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LessorOperatingLeasePaymentsToBeReceivedThreeYears"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceived_1b1f59e1-cfee-488c-bbb2-c9c4dad4ced2" xlink:to="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedThreeYears_b2da1b65-c218-4a77-986c-c289ff107013" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedTwoYears_69a4ca6f-fefa-4cf9-bcec-c4f128d21fba" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LessorOperatingLeasePaymentsToBeReceivedTwoYears"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceived_1b1f59e1-cfee-488c-bbb2-c9c4dad4ced2" xlink:to="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedTwoYears_69a4ca6f-fefa-4cf9-bcec-c4f128d21fba" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedFourYears_9fb40bb8-b384-4805-ae02-217e30dbfcc0" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LessorOperatingLeasePaymentsToBeReceivedFourYears"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceived_1b1f59e1-cfee-488c-bbb2-c9c4dad4ced2" xlink:to="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedFourYears_9fb40bb8-b384-4805-ae02-217e30dbfcc0" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedRemainderOfFiscalYear_be7b0ba9-9d6e-4add-ba42-688cfee7070e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LessorOperatingLeasePaymentsToBeReceivedRemainderOfFiscalYear"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceived_1b1f59e1-cfee-488c-bbb2-c9c4dad4ced2" xlink:to="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedRemainderOfFiscalYear_be7b0ba9-9d6e-4add-ba42-688cfee7070e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_LessorOperatingLeasePaymentToBeReceivedAfterYearFour_3c95cb63-5a4f-4efe-9801-9dfd0072a80b" xlink:href="bhrb-20250630.xsd#bhrb_LessorOperatingLeasePaymentToBeReceivedAfterYearFour"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceived_1b1f59e1-cfee-488c-bbb2-c9c4dad4ced2" xlink:to="loc_bhrb_LessorOperatingLeasePaymentToBeReceivedAfterYearFour_3c95cb63-5a4f-4efe-9801-9dfd0072a80b" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedNextTwelveMonths_f223671d-e382-4948-8f9b-424a6230a312" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LessorOperatingLeasePaymentsToBeReceivedNextTwelveMonths"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceived_1b1f59e1-cfee-488c-bbb2-c9c4dad4ced2" xlink:to="loc_us-gaap_LessorOperatingLeasePaymentsToBeReceivedNextTwelveMonths_f223671d-e382-4948-8f9b-424a6230a312" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofRightofUseAssetsandLiabilitiesbyLeaseTypeDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#LeasedPropertyScheduleofRightofUseAssetsandLiabilitiesbyLeaseTypeDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofRightofUseAssetsandLiabilitiesbyLeaseTypeDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_LeaseRightOfUseAsset_1ba47d2b-7c0e-4110-81df-5beded09ee63" xlink:href="bhrb-20250630.xsd#bhrb_LeaseRightOfUseAsset"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseRightOfUseAsset_927b95ef-5b99-4ed4-bf7c-3375d7526df4" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseRightOfUseAsset"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_LeaseRightOfUseAsset_1ba47d2b-7c0e-4110-81df-5beded09ee63" xlink:to="loc_us-gaap_OperatingLeaseRightOfUseAsset_927b95ef-5b99-4ed4-bf7c-3375d7526df4" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseRightOfUseAsset_dfd8a187-6cec-4926-94ba-21151832c249" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseRightOfUseAsset"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_LeaseRightOfUseAsset_1ba47d2b-7c0e-4110-81df-5beded09ee63" xlink:to="loc_us-gaap_FinanceLeaseRightOfUseAsset_dfd8a187-6cec-4926-94ba-21151832c249" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_LeaseLiability_8cdd53ec-564a-44ab-afe8-abdab0944a4e" xlink:href="bhrb-20250630.xsd#bhrb_LeaseLiability"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseLiability_c82230d7-1784-4993-97a3-2d7853b054f1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseLiability"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_LeaseLiability_8cdd53ec-564a-44ab-afe8-abdab0944a4e" xlink:to="loc_us-gaap_OperatingLeaseLiability_c82230d7-1784-4993-97a3-2d7853b054f1" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiability_a7ab5e8d-b848-4a78-9a76-679e67ed94f4" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiability"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_LeaseLiability_8cdd53ec-564a-44ab-afe8-abdab0944a4e" xlink:to="loc_us-gaap_FinanceLeaseLiability_a7ab5e8d-b848-4a78-9a76-679e67ed94f4" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofLeaseCostDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#LeasedPropertyScheduleofLeaseCostDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofLeaseCostDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LeaseCost_22006965-cdd6-41a9-a1d1-e4fc500d6f26" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LeaseCost"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseRightOfUseAssetAmortization_64497e05-bcd9-413e-bc58-9ac50c7a84e8" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseRightOfUseAssetAmortization"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LeaseCost_22006965-cdd6-41a9-a1d1-e4fc500d6f26" xlink:to="loc_us-gaap_FinanceLeaseRightOfUseAssetAmortization_64497e05-bcd9-413e-bc58-9ac50c7a84e8" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseInterestExpense_06c611c4-735f-413f-a784-0a28cab2f792" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseInterestExpense"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LeaseCost_22006965-cdd6-41a9-a1d1-e4fc500d6f26" xlink:to="loc_us-gaap_FinanceLeaseInterestExpense_06c611c4-735f-413f-a784-0a28cab2f792" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseCost_e2409d72-0a59-43df-874b-e1e4ef25466d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseCost"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LeaseCost_22006965-cdd6-41a9-a1d1-e4fc500d6f26" xlink:to="loc_us-gaap_OperatingLeaseCost_e2409d72-0a59-43df-874b-e1e4ef25466d" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofLesseeOperatingLeaseandFinancingLeasePaymenttobeReceivedFiscalYearMaturityDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#LeasedPropertyScheduleofLesseeOperatingLeaseandFinancingLeasePaymenttobeReceivedFiscalYearMaturityDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofLesseeOperatingLeaseandFinancingLeasePaymenttobeReceivedFiscalYearMaturityDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_2ca9b8e8-f635-46cd-aa23-1bf77be3d345" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiabilityPaymentsDue"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiability_a3faa482-7cf7-4fac-865c-5e9783a349c2" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiability"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_2ca9b8e8-f635-46cd-aa23-1bf77be3d345" xlink:to="loc_us-gaap_FinanceLeaseLiability_a3faa482-7cf7-4fac-865c-5e9783a349c2" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiabilityUndiscountedExcessAmount_3da07ccb-1846-45f2-8311-6575aedf4889" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiabilityUndiscountedExcessAmount"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_2ca9b8e8-f635-46cd-aa23-1bf77be3d345" xlink:to="loc_us-gaap_FinanceLeaseLiabilityUndiscountedExcessAmount_3da07ccb-1846-45f2-8311-6575aedf4889" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_44d7dfa5-44da-4052-b10a-55ebdc578020" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseLiability_0a7d6780-46c3-4c12-979c-bc44e7226170" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseLiability"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_44d7dfa5-44da-4052-b10a-55ebdc578020" xlink:to="loc_us-gaap_OperatingLeaseLiability_0a7d6780-46c3-4c12-979c-bc44e7226170" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount_7250e28c-09a7-46f2-be1b-2029cf425e8d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_44d7dfa5-44da-4052-b10a-55ebdc578020" xlink:to="loc_us-gaap_LesseeOperatingLeaseLiabilityUndiscountedExcessAmount_7250e28c-09a7-46f2-be1b-2029cf425e8d" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofLesseeOperatingLeaseandFinancingLeasePaymenttobeReceivedFiscalYearMaturityDetails_1" xlink:type="simple" xlink:href="bhrb-20250630.xsd#LeasedPropertyScheduleofLesseeOperatingLeaseandFinancingLeasePaymenttobeReceivedFiscalYearMaturityDetails_1"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/LeasedPropertyScheduleofLesseeOperatingLeaseandFinancingLeasePaymenttobeReceivedFiscalYearMaturityDetails_1" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_ed072b8b-a8de-4b76-b452-226bb69c7748" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear_922d0bfd-c949-4d2e-bfff-2a1d91d851b3" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_ed072b8b-a8de-4b76-b452-226bb69c7748" xlink:to="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsRemainderOfFiscalYear_922d0bfd-c949-4d2e-bfff-2a1d91d851b3" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_09fa8147-9b01-4725-be58-ebd140460c2e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_ed072b8b-a8de-4b76-b452-226bb69c7748" xlink:to="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_09fa8147-9b01-4725-be58-ebd140460c2e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_1a799d14-41d1-4c75-9842-eb2d0669d34b" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_ed072b8b-a8de-4b76-b452-226bb69c7748" xlink:to="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_1a799d14-41d1-4c75-9842-eb2d0669d34b" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_LesseeOperatingLeaseLiabilityToBePaidDueAfterYearFour_abd273fe-e53d-4f7f-a2cf-47eeed04c59f" xlink:href="bhrb-20250630.xsd#bhrb_LesseeOperatingLeaseLiabilityToBePaidDueAfterYearFour"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_ed072b8b-a8de-4b76-b452-226bb69c7748" xlink:to="loc_bhrb_LesseeOperatingLeaseLiabilityToBePaidDueAfterYearFour_abd273fe-e53d-4f7f-a2cf-47eeed04c59f" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_ffce187d-a86e-4e00-a355-ea21722af458" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_ed072b8b-a8de-4b76-b452-226bb69c7748" xlink:to="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_ffce187d-a86e-4e00-a355-ea21722af458" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_8927b934-82d4-4530-837e-41e275baa15a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDue_ed072b8b-a8de-4b76-b452-226bb69c7748" xlink:to="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_8927b934-82d4-4530-837e-41e275baa15a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_45600f83-840d-4205-8c0d-8610640d4dff" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiabilityPaymentsDue"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiabilityPaymentsDueYearThree_e3a3e3ce-46e8-4463-964f-d964edbb2d67" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiabilityPaymentsDueYearThree"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_45600f83-840d-4205-8c0d-8610640d4dff" xlink:to="loc_us-gaap_FinanceLeaseLiabilityPaymentsDueYearThree_e3a3e3ce-46e8-4463-964f-d964edbb2d67" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_LessorFinancingLeasePaymentToBeReceivedAfterYearFour_7e024e49-20a4-4bb2-9ee2-e019e0abaf41" xlink:href="bhrb-20250630.xsd#bhrb_LessorFinancingLeasePaymentToBeReceivedAfterYearFour"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_45600f83-840d-4205-8c0d-8610640d4dff" xlink:to="loc_bhrb_LessorFinancingLeasePaymentToBeReceivedAfterYearFour_7e024e49-20a4-4bb2-9ee2-e019e0abaf41" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiabilityPaymentsDueYearTwo_874030f1-285f-4f1d-9bfc-f036922b076a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiabilityPaymentsDueYearTwo"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_45600f83-840d-4205-8c0d-8610640d4dff" xlink:to="loc_us-gaap_FinanceLeaseLiabilityPaymentsDueYearTwo_874030f1-285f-4f1d-9bfc-f036922b076a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiabilityPaymentsRemainderOfFiscalYear_3f52abc4-3e00-431b-a16d-7ceaf585273a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiabilityPaymentsRemainderOfFiscalYear"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_45600f83-840d-4205-8c0d-8610640d4dff" xlink:to="loc_us-gaap_FinanceLeaseLiabilityPaymentsRemainderOfFiscalYear_3f52abc4-3e00-431b-a16d-7ceaf585273a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiabilityPaymentsDueYearFour_47688d67-e033-4806-bf1a-18f3518db21e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiabilityPaymentsDueYearFour"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_45600f83-840d-4205-8c0d-8610640d4dff" xlink:to="loc_us-gaap_FinanceLeaseLiabilityPaymentsDueYearFour_47688d67-e033-4806-bf1a-18f3518db21e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinanceLeaseLiabilityPaymentsDueNextTwelveMonths_6bfd1dd4-e0b2-4578-92b5-ed9a68b5762a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinanceLeaseLiabilityPaymentsDueNextTwelveMonths"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FinanceLeaseLiabilityPaymentsDue_45600f83-840d-4205-8c0d-8610640d4dff" xlink:to="loc_us-gaap_FinanceLeaseLiabilityPaymentsDueNextTwelveMonths_6bfd1dd4-e0b2-4578-92b5-ed9a68b5762a" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/OtherOperatingExpenseScheduleofOtherOperatingExpenseDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#OtherOperatingExpenseScheduleofOtherOperatingExpenseDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/OtherOperatingExpenseScheduleofOtherOperatingExpenseDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherNoninterestExpense"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_DonationExpense_f83709f7-2b57-49e5-a866-ee2c1cbe4d52" xlink:href="bhrb-20250630.xsd#bhrb_DonationExpense"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_bhrb_DonationExpense_f83709f7-2b57-49e5-a866-ee2c1cbe4d52" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FederalDepositInsuranceCorporationPremiumExpense_0847a1f0-c04d-4990-b4a4-0b6a9617853e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FederalDepositInsuranceCorporationPremiumExpense"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_FederalDepositInsuranceCorporationPremiumExpense_0847a1f0-c04d-4990-b4a4-0b6a9617853e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_ConsultantFees_52186e24-f1a4-48e1-8283-a0e06c380e98" xlink:href="bhrb-20250630.xsd#bhrb_ConsultantFees"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_bhrb_ConsultantFees_52186e24-f1a4-48e1-8283-a0e06c380e98" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LegalFees_59e61649-b8d6-4f4c-bc48-2c60a393a8c6" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LegalFees"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_LegalFees_59e61649-b8d6-4f4c-bc48-2c60a393a8c6" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AmortizationOfIntangibleAssets_12989121-ed55-4c62-97a6-0922604c3899" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AmortizationOfIntangibleAssets"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_AmortizationOfIntangibleAssets_12989121-ed55-4c62-97a6-0922604c3899" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProfessionalFees_c345e6a4-7d6c-4fda-bc6b-3fb76abd468a" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProfessionalFees"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_ProfessionalFees_c345e6a4-7d6c-4fda-bc6b-3fb76abd468a" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_ATMCardAndNetworkExpense_0f657e8d-5923-4d96-89be-08be938027d7" xlink:href="bhrb-20250630.xsd#bhrb_ATMCardAndNetworkExpense"/>
    <link:calculationArc order="7" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_bhrb_ATMCardAndNetworkExpense_0f657e8d-5923-4d96-89be-08be938027d7" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NoninterestExpenseDirectorsFees_bc1e9923-0d37-44bc-8aec-185cf180df6e" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NoninterestExpenseDirectorsFees"/>
    <link:calculationArc order="8" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_NoninterestExpenseDirectorsFees_bc1e9923-0d37-44bc-8aec-185cf180df6e" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InformationTechnologyAndDataProcessing_a130b51b-7fea-4a75-afbb-92b96a473ecf" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InformationTechnologyAndDataProcessing"/>
    <link:calculationArc order="9" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_InformationTechnologyAndDataProcessing_a130b51b-7fea-4a75-afbb-92b96a473ecf" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TaxesExcludingIncomeAndExciseTaxes_9e536532-134b-4bc6-af13-c161141edb3c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TaxesExcludingIncomeAndExciseTaxes"/>
    <link:calculationArc order="10" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_TaxesExcludingIncomeAndExciseTaxes_9e536532-134b-4bc6-af13-c161141edb3c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_MarketingExpense_8f2c6107-158e-4d24-bfc4-a8e2b7ad7763" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_MarketingExpense"/>
    <link:calculationArc order="11" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_MarketingExpense_8f2c6107-158e-4d24-bfc4-a8e2b7ad7763" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherExpenses_d5f8e49f-4229-4768-9ebb-d1fbb680e6cd" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherExpenses"/>
    <link:calculationArc order="12" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_us-gaap_OtherExpenses_d5f8e49f-4229-4768-9ebb-d1fbb680e6cd" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_AmortizationOfHistoricTaxCredit_d4a76e1a-8d60-492c-bf0d-d3743549b0ae" xlink:href="bhrb-20250630.xsd#bhrb_AmortizationOfHistoricTaxCredit"/>
    <link:calculationArc order="13" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_OtherNoninterestExpense_4b1e46b6-9321-4ec8-8b9f-9b36b3a06d1d" xlink:to="loc_bhrb_AmortizationOfHistoricTaxCredit_d4a76e1a-8d60-492c-bf0d-d3743549b0ae" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/EarningsPerShareScheduleofEarningsPerShareBasicandDilutedDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#EarningsPerShareScheduleofEarningsPerShareBasicandDilutedDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/EarningsPerShareScheduleofEarningsPerShareBasicandDilutedDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_2bca815d-a26b-4fad-8d89-594e2956a678" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncrementalCommonSharesAttributableToShareBasedPaymentArrangements_123681f5-951f-4b06-9334-732f33908fb2" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncrementalCommonSharesAttributableToShareBasedPaymentArrangements"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_2bca815d-a26b-4fad-8d89-594e2956a678" xlink:to="loc_us-gaap_IncrementalCommonSharesAttributableToShareBasedPaymentArrangements_123681f5-951f-4b06-9334-732f33908fb2" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_7238b070-617f-46d2-bd55-9c1dc56cbcb5" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_WeightedAverageNumberOfSharesOutstandingBasic"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_WeightedAverageNumberOfDilutedSharesOutstanding_2bca815d-a26b-4fad-8d89-594e2956a678" xlink:to="loc_us-gaap_WeightedAverageNumberOfSharesOutstandingBasic_7238b070-617f-46d2-bd55-9c1dc56cbcb5" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/BusinessCombinationScheduleofAssetsAcquiredandLiabilitiesAssumedDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#BusinessCombinationScheduleofAssetsAcquiredandLiabilitiesAssumedDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/BusinessCombinationScheduleofAssetsAcquiredandLiabilitiesAssumedDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities_67330e18-b2fa-4c7f-b6e8-ce9469c6c0de" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteDeposits_50e0a498-0063-40b0-9062-86bcb18c7352" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteDeposits"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities_67330e18-b2fa-4c7f-b6e8-ce9469c6c0de" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteDeposits_50e0a498-0063-40b0-9062-86bcb18c7352" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteSubordinatedDebenturesAndTrustPreferredSecurities_4dc39a01-7b8e-410d-a4a8-82e2df9e0600" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteSubordinatedDebenturesAndTrustPreferredSecurities"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities_67330e18-b2fa-4c7f-b6e8-ce9469c6c0de" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteSubordinatedDebenturesAndTrustPreferredSecurities_4dc39a01-7b8e-410d-a4a8-82e2df9e0600" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteUnfundedReserveLiability_6babb7b0-9d92-4c6a-92f0-6329c6deb7ea" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteUnfundedReserveLiability"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities_67330e18-b2fa-4c7f-b6e8-ce9469c6c0de" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteUnfundedReserveLiability_6babb7b0-9d92-4c6a-92f0-6329c6deb7ea" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAccruedInterestAndOtherLiabilities_750af689-a6b2-467b-a83b-658766ceb0a1" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAccruedInterestAndOtherLiabilities"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities_67330e18-b2fa-4c7f-b6e8-ce9469c6c0de" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAccruedInterestAndOtherLiabilities_750af689-a6b2-467b-a83b-658766ceb0a1" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill_a1cbdb72-e596-442e-8ff8-469f6d97b540" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentPropertyPlantAndEquipment_3bd338b5-cb98-4b1c-96c1-e575356fdabb" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentPropertyPlantAndEquipment"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill_a1cbdb72-e596-442e-8ff8-469f6d97b540" xlink:to="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentPropertyPlantAndEquipment_3bd338b5-cb98-4b1c-96c1-e575356fdabb" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_ef60d764-eea0-408f-8816-fbc3eab1fcb3" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill_a1cbdb72-e596-442e-8ff8-469f6d97b540" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_ef60d764-eea0-408f-8816-fbc3eab1fcb3" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentOtherNoncurrentAssets_6ee759b0-3ff3-400a-ae3c-dbdcbd3ff448" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentOtherNoncurrentAssets"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill_a1cbdb72-e596-442e-8ff8-469f6d97b540" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentOtherNoncurrentAssets_6ee759b0-3ff3-400a-ae3c-dbdcbd3ff448" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentGoodwillAndIntangibles_5bceeca1-1361-4d8f-b5c4-bd31b251838f" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentGoodwillAndIntangibles"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill_a1cbdb72-e596-442e-8ff8-469f6d97b540" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentGoodwillAndIntangibles_5bceeca1-1361-4d8f-b5c4-bd31b251838f" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentDebtSecuritiesHeldToMaturityAtAmortizedCost_96cc97ed-ec7f-4693-8427-357f28f89dcb" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentDebtSecuritiesHeldToMaturityAtAmortizedCost"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill_a1cbdb72-e596-442e-8ff8-469f6d97b540" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentDebtSecuritiesHeldToMaturityAtAmortizedCost_96cc97ed-ec7f-4693-8427-357f28f89dcb" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_d48844cb-a38f-400d-ade0-455dd247bad3" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableAllowanceForCreditLoss_caccacb0-9b79-423c-9df3-ae4650f17b02" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableAllowanceForCreditLoss"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_d48844cb-a38f-400d-ade0-455dd247bad3" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableAllowanceForCreditLoss_caccacb0-9b79-423c-9df3-ae4650f17b02" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentCurrentAssetsReceivables_e25900b0-bc23-40d0-a992-d828327906ac" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentCurrentAssetsReceivables"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_d48844cb-a38f-400d-ade0-455dd247bad3" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentCurrentAssetsReceivables_e25900b0-bc23-40d0-a992-d828327906ac" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentConsiderationTransferred_e96a05cc-f8c0-4d24-b4e8-97d2b4ad2511" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentConsiderationTransferred"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities_7ae76896-be3f-4cbc-a4b7-72a76c73ee6b" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentConsiderationTransferred_e96a05cc-f8c0-4d24-b4e8-97d2b4ad2511" xlink:to="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentFinancialLiabilities_7ae76896-be3f-4cbc-a4b7-72a76c73ee6b" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill_275870db-5ce7-4f05-992f-d8c92afd74e1" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAdjustmentConsiderationTransferred_e96a05cc-f8c0-4d24-b4e8-97d2b4ad2511" xlink:to="loc_bhrb_BusinessCombinationProvisionalInformationInitialAccountingIncompleteAssetsAndGoodwill_275870db-5ce7-4f05-992f-d8c92afd74e1" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_06f329d1-2cca-44b6-a3a1-13e0a59bdbab" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableAllowanceForCreditLoss_a80df0bf-1dcf-4614-93a5-41ca8b10c467" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableAllowanceForCreditLoss"/>
    <link:calculationArc order="1" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_06f329d1-2cca-44b6-a3a1-13e0a59bdbab" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableAllowanceForCreditLoss_a80df0bf-1dcf-4614-93a5-41ca8b10c467" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsReceivables_9d11d451-1896-49ac-8df8-b54eac44d6f3" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsReceivables"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_06f329d1-2cca-44b6-a3a1-13e0a59bdbab" xlink:to="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsReceivables_9d11d451-1896-49ac-8df8-b54eac44d6f3" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDebtSecuritiesHeldToMaturityAtAmortizedCost_349c057e-6945-49d3-a7da-fc1f2395d627" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDebtSecuritiesHeldToMaturityAtAmortizedCost"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDebtSecuritiesHeldToMaturityAtAmortizedCost_349c057e-6945-49d3-a7da-fc1f2395d627" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedInterestReceivable_df3f1ed7-79f2-4cd0-a231-b31489afa013" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedInterestReceivable"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedInterestReceivable_df3f1ed7-79f2-4cd0-a231-b31489afa013" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsDebtSecuritiesAvailableForSale_69126c75-213b-4cb2-ae14-774ac6ee30b0" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsDebtSecuritiesAvailableForSale"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsDebtSecuritiesAvailableForSale_69126c75-213b-4cb2-ae14-774ac6ee30b0" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_71581dee-44ae-4ca9-9af0-b8fe3e71cddb" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFinancingReceivableExcludingAccruedInterestAfterAllowanceForCreditLoss_71581dee-44ae-4ca9-9af0-b8fe3e71cddb" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityAndOtherInvestments_da559f32-9caf-4140-9812-abd84f63eddd" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityAndOtherInvestments"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityAndOtherInvestments_da559f32-9caf-4140-9812-abd84f63eddd" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedPropertyPlantAndEquipment_ca611a6a-95c7-4279-a3c8-385c2eeab2dd" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedPropertyPlantAndEquipment"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedPropertyPlantAndEquipment_ca611a6a-95c7-4279-a3c8-385c2eeab2dd" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_fec5f9e0-84fc-433c-94c8-5410d6119923" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents"/>
    <link:calculationArc order="7" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_fec5f9e0-84fc-433c-94c8-5410d6119923" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedGoodwillAndIntangibleAssets_aae3aed0-23ad-4595-a220-8e8aba94eed6" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedGoodwillAndIntangibleAssets"/>
    <link:calculationArc order="8" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedGoodwillAndIntangibleAssets_aae3aed0-23ad-4595-a220-8e8aba94eed6" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedOtherNoncurrentAssets_5ecbc82e-0d1f-4340-89a5-2278a1bbc502" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedOtherNoncurrentAssets"/>
    <link:calculationArc order="9" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedOtherNoncurrentAssets_5ecbc82e-0d1f-4340-89a5-2278a1bbc502" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCompanyOwnedLifeInsurance_f1917fc0-04a9-495a-ae23-9d3e1fb9a124" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCompanyOwnedLifeInsurance"/>
    <link:calculationArc order="10" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_c4ac79db-be45-46ef-b7c9-da25f759c16f" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCompanyOwnedLifeInsurance_f1917fc0-04a9-495a-ae23-9d3e1fb9a124" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredGoodwillAndLiabilitiesAssumedNet_eba9e9b3-6c1f-45e0-a150-5f68ada19026" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredGoodwillAndLiabilitiesAssumedNet"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_249c5c16-5900-41e0-95ef-65df864a696f" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredGoodwillAndLiabilitiesAssumedNet_eba9e9b3-6c1f-45e0-a150-5f68ada19026" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAssetsAndGoodwill_249c5c16-5900-41e0-95ef-65df864a696f" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_94601a21-604c-4c80-a647-31953a2b8f17" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities"/>
    <link:calculationArc order="2" weight="-1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredGoodwillAndLiabilitiesAssumedNet_eba9e9b3-6c1f-45e0-a150-5f68ada19026" xlink:to="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_94601a21-604c-4c80-a647-31953a2b8f17" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_e6153019-e830-41b1-a0be-0d9f37f323c2" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDeposits_17533868-d473-4373-82b6-be87dc021ceb" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDeposits"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_e6153019-e830-41b1-a0be-0d9f37f323c2" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedDeposits_17533868-d473-4373-82b6-be87dc021ceb" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedInterestAndOtherLiabilities_21005d5a-83d4-40c8-b081-c79e062d6069" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedInterestAndOtherLiabilities"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_e6153019-e830-41b1-a0be-0d9f37f323c2" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedInterestAndOtherLiabilities_21005d5a-83d4-40c8-b081-c79e062d6069" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedSubordinatedDebenturesAndTrustPreferredSecurities_cf120063-d4fc-4e3e-b923-a3c9f50ab81d" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedSubordinatedDebenturesAndTrustPreferredSecurities"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_e6153019-e830-41b1-a0be-0d9f37f323c2" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedSubordinatedDebenturesAndTrustPreferredSecurities_cf120063-d4fc-4e3e-b923-a3c9f50ab81d" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedUnfundedReserveLiability_d107f60e-258c-44d5-a283-e19fc6e25cd6" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedUnfundedReserveLiability"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_e6153019-e830-41b1-a0be-0d9f37f323c2" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedUnfundedReserveLiability_d107f60e-258c-44d5-a283-e19fc6e25cd6" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedBorrowings_cc2c551f-a649-4bd9-b18c-c47264202d02" xlink:href="bhrb-20250630.xsd#bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedBorrowings"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilities_e6153019-e830-41b1-a0be-0d9f37f323c2" xlink:to="loc_bhrb_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedBorrowings_cc2c551f-a649-4bd9-b18c-c47264202d02" xlink:type="arc"/>
  </link:calculationLink>
  <link:roleRef roleURI="http://www.burkeandherbertbank.com/role/GoodwillandOtherIntangibleAssetsScheduleofFiniteLivedIntangibleAssetsFutureAmortizationExpenseDetails" xlink:type="simple" xlink:href="bhrb-20250630.xsd#GoodwillandOtherIntangibleAssetsScheduleofFiniteLivedIntangibleAssetsFutureAmortizationExpenseDetails"/>
  <link:calculationLink xlink:role="http://www.burkeandherbertbank.com/role/GoodwillandOtherIntangibleAssetsScheduleofFiniteLivedIntangibleAssetsFutureAmortizationExpenseDetails" xlink:type="extended">
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsNet_ab224031-190e-4da6-827a-b1d481c45cba" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsNet"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearThree_b199547b-d259-4c5b-9261-40e4da5a435c" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearThree"/>
    <link:calculationArc order="1" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsNet_ab224031-190e-4da6-827a-b1d481c45cba" xlink:to="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearThree_b199547b-d259-4c5b-9261-40e4da5a435c" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRemainderOfFiscalYear_a70306fd-be36-48b6-a938-0f97d0e514bb" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRemainderOfFiscalYear"/>
    <link:calculationArc order="2" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsNet_ab224031-190e-4da6-827a-b1d481c45cba" xlink:to="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseRemainderOfFiscalYear_a70306fd-be36-48b6-a938-0f97d0e514bb" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearFour_76b27280-24e8-4bfd-b66a-5d6eea4bbd8d" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearFour"/>
    <link:calculationArc order="3" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsNet_ab224031-190e-4da6-827a-b1d481c45cba" xlink:to="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearFour_76b27280-24e8-4bfd-b66a-5d6eea4bbd8d" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths_867009b3-e121-4649-98be-a8e1c2282816" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths"/>
    <link:calculationArc order="4" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsNet_ab224031-190e-4da6-827a-b1d481c45cba" xlink:to="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseNextTwelveMonths_867009b3-e121-4649-98be-a8e1c2282816" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo_187c1967-19c1-43e6-8e07-43865779e59b" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo"/>
    <link:calculationArc order="5" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsNet_ab224031-190e-4da6-827a-b1d481c45cba" xlink:to="loc_us-gaap_FiniteLivedIntangibleAssetsAmortizationExpenseYearTwo_187c1967-19c1-43e6-8e07-43865779e59b" xlink:type="arc"/>
    <link:loc xlink:type="locator" xlink:label="loc_bhrb_FiniteLivedIntangibleAssetExpectedAmortizationAfterYearFour_2ef5feea-24a1-43cd-a734-0992a4c4228c" xlink:href="bhrb-20250630.xsd#bhrb_FiniteLivedIntangibleAssetExpectedAmortizationAfterYearFour"/>
    <link:calculationArc order="6" weight="1.0" xlink:arcrole="http://www.xbrl.org/2003/arcrole/summation-item" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsNet_ab224031-190e-4da6-827a-b1d481c45cba" xlink:to="loc_bhrb_FiniteLivedIntangibleAssetExpectedAmortizationAfterYearFour_2ef5feea-24a1-43cd-a734-0992a4c4228c" xlink:type="arc"/>
  </link:calculationLink>
</link:linkbase>
