here is the actual state law regarding amending pistol permits.
9. License: amendment. Elsewhere than in the city of New York, a person licensed to carry or possess a pistol or revolver may apply at any time to his licensing officer for amendment of his license to include one or more such weapons or to cancel weapons held under license. If granted, a record of the amendment describing the weapons involved shall be filed by the licensing officer in the executive department, division of state police, Albany. Notification of any change of residence shall be made in writing by any licensee within ten days after such change occurs, and a record of such change shall be inscribed by such licensee on the reverse side of his license. Elsewhere than in the city of New York, and in the counties of Nassau and Suffolk, such notification shall be made to the executive department, division of state police, Albany, and in the city of New York to the police commissioner of that city, and in the county of Nassau to the police commissioner of that county, and in the county of Suffolk to the licensing officer of that county, who shall, within ten days after such notification shall be received by him, give notice in writing of such change to the executive department, division of state police, at Albany.