N.Y. Dom. Rel. Law § 20
All original affidavits, statements, consents and licenses with certificates attached, and also all written contracts of marriages outside of the city of New York shall be kept on file and properly indexed by the state department of health, and such similar evidences of marriage in the city of New York shall be kept on file and properly indexed by the city clerk of the city of New York. They shall be carefully examined, and if any such are incomplete or unsatisfactory the state commissioner of health and in the city of New York the city clerk shall require such further information to be supplied as may be necessary to make the record complete and satisfactory. Whenever it is claimed that a mistake has been made through inadvertence in any of the statements, affidavits or other papers required by this section to be filed with the state department of health, and in the city of New York with the city clerk's office, the state commissioner of health and in the city of New York the city clerk may file with the same, affidavits upon the part of the person claiming to be aggrieved by such mistake, showing the true facts and the reason for the mistake and may make a note upon such original paper, statement or affidavit showing that a mistake is claimed to have been made and the nature thereof.
N.Y. Dom. Rel.Law § 20