N.Y. Elec. Law § 8-314
If a registered voter marked an enrollment blank, but his enrollment as so indicated was not entered or entered incorrectly, he nevertheless shall be permitted to vote at the primary election, with the party under whose name he marked such blank, if he produces a certificate of the board of elections that he marked an enrollment blank, specifying the party, and that the entry of his enrollment on the registration poll record was omitted or incorrectly entered by inadvertence or mistake and he is qualified to vote at the primary election in the party set forth in such certificate. In such a case the board of elections shall issue such certificate on demand. The inspectors shall then and there enter the enrollment on the record.
N.Y. Elec. Law § 8-314