Exhibit 5.1

Kilpatrick Townsend & Stockton LLP   701 Pennsylvania Avenue, NW, Suite 200 Washington, DC 20004 202.508.5800   www.ktslaw.com

June 4, 2026

Board of Directors

Hanover Bancorp, Inc.

80 East Jericho Turnpike

Mineola, New York 11501

Ladies and Gentlemen:

We have acted as counsel to Hanover Bancorp, Inc., a Maryland corporation (the “Company”), in connection with the preparation and filing with the Securities and Exchange Commission (the “Commission”) pursuant to the Securities Act of 1933, as amended (“Securities Act”), of the registration statement on Form S-4 (the “Registration Statement”) that was filed by the Company with the Commission.

The Registration Statement relates to the offering by the Company of up to $35,000,000 in aggregate principal amount of the Company’s 7.25% Fixed-to-Floating Rate Subordinated Notes due 2036 (the “New Notes”) in exchange for a like principal amount of the Company’s outstanding unregistered 7.25% Fixed-to-Floating Rate Subordinated Notes due 2036 (the “Old Notes”). The New Notes will be issued pursuant to the Indenture, dated as of March 12, 2026 (the “Indenture”), between the Company and UMB Bank, N.A., as trustee (the “Trustee”), as contemplated by the Registration Rights Agreements, dated as of March 12, 2026, by and among the Company and the purchasers of the Old Notes (the “Registration Rights Agreements”).

In connection with the issuance of this opinion letter, we have examined originals or copies, certified or otherwise identified to our satisfaction, of: (i) the Registration Statement and all exhibits thereto; (ii) the Old Notes; (iii) the New Notes; (iv) the Indenture; (v) the Registration Rights Agreements; (vi) the articles of incorporation and bylaws of the Company; and (vii) certain resolutions adopted by the Board of Directors of the Company relating to the exchange offer, the issuance of the New Notes and the Indenture. We have also examined the originals or duplicates or certified or conformed copies of such corporate records, agreements, documents and other instruments, as we have deemed necessary, and have made such other investigations as we have deemed necessary or appropriate to enable us to render this opinion.

Board of Directors

Hanover Bancorp, Inc.

June 4, 2026

Page 2

We have examined and relied on originals or copies certified or otherwise identified to our satisfaction of such documents, corporate records, certificates of public officials and other instruments as we have deemed necessary or appropriate for the purposes of this opinion. In such examination, we have assumed: (i) the authenticity of original documents and the genuineness of all signatures (whether manual, electronic or otherwise) and, to the extent that a signature on a document is manifested by electronic or similar means, such signature has been executed or adopted by a signatory with an intent to authenticate and sign the document; (ii) the persons executing the documents we have examined have the legal capacity to execute such documents; (iii) the conformity to the originals of all documents submitted to us as copies; (iv) the truth, accuracy and completeness of the information, representations and warranties contained in the Registration Statement and other records, documents, instruments and certificates we have reviewed; (v) all New Notes will be issued in the manner stated in the Registration Rights Agreements and the Registration Statement; (vi) the Indenture has been duly authorized, executed and delivered by the Trustee, constitutes the valid, binding and enforceable obligation of the Trustee, and the Trustee is duly organized, validly existing and in good standing and has full power and authority to act as trustee under the Indenture; and (vii) the execution and delivery by the Company of the New Notes and the performance by the Company of its obligations thereunder do not and will not violate, conflict with or constitute a default under any agreement or instrument to which the Company or its properties is subject (other than those agreements and instruments that have been identified to us by the Company and which are listed in Part II of the Registration Statement as exhibits thereto). As to any facts material to the opinions expressed herein, we have relied upon statements and representations of officers and other representatives of the Company and others.

Based upon the foregoing, and subject to the qualifications, assumptions and limitations stated herein, we are of the opinion that when the New Notes (in the form examined by us) are duly executed by the Company, authenticated by the Trustee in accordance with the Indenture and issued and delivered upon consummation of the exchange offer (as described in the Registration Statement) against receipt of the Old Notes surrendered in exchange therefor in accordance with the terms of such exchange offer, the Registration Rights Agreements, the Registration Statement and the Indenture, the New Notes will constitute valid and binding obligations of the Company, enforceable against the Company in accordance with their terms.

The opinion is limited in all respects to the laws of the State of Maryland and the laws of the State of New York and we express no opinion as to federal law, including the federal securities laws, state securities (or “blue sky” laws) or the laws of any other jurisdiction. Our opinions expressed herein are as of the date hereof, and we assume no obligation to revise or supplement the opinions rendered herein should the above-referenced laws be changed by legislative or regulatory action, judicial decision or otherwise.

Our opinion is subject to: (1) applicable bankruptcy, insolvency, reorganization, receivership, conservatorship, fraudulent conveyance, moratorium or other similar laws relating to or affecting the rights of creditors generally; (2) general principles of equity, including without limitation concepts of materiality, reasonableness, good faith and fair dealing, and the possible unavailability of specific performance or injunctive relief and limitation of rights of acceleration, regardless of whether such enforceability is considered in a proceeding in equity or at law; and (3) limitation of rights to indemnification, exculpation and contribution, which may be limited by applicable law or equitable principles. We express no opinion regarding the effectiveness of any waiver of stay, extension or usury laws or of unknown future rights, and we express no opinion regarding severability provisions.

Board of Directors

Hanover Bancorp, Inc.

June 4, 2026

Page 3

This opinion letter has been prepared, and is to be understood, in accordance with customary practice of lawyers who regularly give and lawyers who regularly advise recipients regarding opinions of this kind, is limited to the matters expressly stated herein and is provided solely in connection with the transactions contemplated by the Registration Statement and no opinions may be inferred or implied beyond the matters expressly stated herein. The opinion expressed herein is rendered and speaks only as of the date hereof and we specifically disclaim any responsibility to update such opinion subsequent to the date hereof or to advise you of subsequent developments affecting such opinion.

We hereby consent to the filing of this opinion letter as Exhibit 5.1 to the Registration Statement and to the reference to us under the heading “Legal Matters” in the prospectus forming part of the Registration Statement and any amendments and supplements thereto. In giving such consent, we do not thereby admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act or the rules and regulations promulgated thereunder.