N.Y. Elec. Law § 5-403
If the ballot of any person, voted in an affidavit envelope in the manner prescribed by this chapter, is rejected under the provisions of this chapter on the grounds that such person is not a qualified voter of the election district wherein he or she sought to vote, or is not duly enrolled in the party in whose primary he or she sought to vote and if such person has completed the application for registration and enrollment or change of enrollment on such affidavit envelope, the board of elections shall process such an application in the same manner as other applications for registration and enrollment or change of enrollment. The board of elections shall immediately notify such person by first class mail directed to the address given in his or her affidavit, of the rejection of his or her ballot, together with the reason therefor and, if such person has not completed the application for registration by mail, the appropriate information on the times and places where he or she may register, re-register, enroll, or change or correct his or her enrollment. Where appropriate, a form of application for personal registration by mail shall be included with such notice.
N.Y. Elec. Law § 5-403