<?xml version="1.0"?>
<ownershipDocument>

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    <periodOfReport>2021-05-31</periodOfReport>

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    <issuer>
        <issuerCik>0001580156</issuerCik>
        <issuerName>Houghton Mifflin Harcourt Co</issuerName>
        <issuerTradingSymbol>HMHC</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001353671</rptOwnerCik>
            <rptOwnerName>Greenthal Jill A</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O HOUGHTON MIFFLIN HARCOURT CO.</rptOwnerStreet1>
            <rptOwnerStreet2>125 HIGH STREET</rptOwnerStreet2>
            <rptOwnerCity>BOSTON</rptOwnerCity>
            <rptOwnerState>MA</rptOwnerState>
            <rptOwnerZipCode>02110</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

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                <equitySwapInvolved>0</equitySwapInvolved>
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            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
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                <transactionPricePerShare>
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                    <footnoteId id="F1"/>
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            </transactionAmounts>
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                </sharesOwnedFollowingTransaction>
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            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
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            <expirationDate>
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                <underlyingSecurityShares>
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                </underlyingSecurityShares>
            </underlyingSecurity>
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                </directOrIndirectOwnership>
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        <derivativeTransaction>
            <securityTitle>
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                <value></value>
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                <transactionShares>
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                </transactionShares>
                <transactionPricePerShare>
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                <value></value>
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            <transactionAmounts>
                <transactionShares>
                    <value>55556</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
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                    <value>55556</value>
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                </sharesOwnedFollowingTransaction>
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                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
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        </derivativeTransaction>
        <derivativeTransaction>
            <securityTitle>
                <value>Phantom Stock</value>
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                <footnoteId id="F1"/>
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            <transactionDate>
                <value>2021-05-31</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
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                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
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            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>55556</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
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            <exerciseDate>
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            <expirationDate>
                <footnoteId id="F4"/>
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            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>55556</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>55556</value>
                </sharesOwnedFollowingTransaction>
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            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
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        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Each restricted stock unit and each share of phantom stock represents a contingent right to receive one share of Houghton Mifflin Harcourt Company's (the &quot;Company&quot;) common stock.</footnote>
        <footnote id="F2">The shares of phantom stock were settled in exchange for a like amount of shares of the Company's common stock.</footnote>
        <footnote id="F3">This grant of restricted stock units will vest on May 31, 2022, subject to continued service on the Company's board of directors through such vesting date, with settlement of the restricted stock units to be deferred until May 31, 2024 (or, upon separation from service after the vesting date, such earlier settlement date determined in accordance with the Company's Non-Employee Director Deferred Compensation Plan).</footnote>
        <footnote id="F4">Upon the vesting of the restricted stock units granted to the reporting person on May 31, 2020, the reporting person deferred the receipt of 55,556 shares of common stock and received instead 55,556 shares of phantom stock pursuant to the Company's Non-Employee Director Deferred Compensation Plan. The phantom stock becomes payable on May 31, 2023 (or, upon separation from service, at such earlier settlement date determined in accordance with the Company's Non-Employee Director Deferred Compensation Plan).</footnote>
    </footnotes>

    <remarks>William F. Bayers is the Executive Vice President, Secretary and General Counsel of the Company.</remarks>

    <ownerSignature>
        <signatureName>/s/ William F. Bayers, Attorney-in-Fact</signatureName>
        <signatureDate>2021-06-02</signatureDate>
    </ownerSignature>
</ownershipDocument>
