N.Y. Second Class Cities Law § 54
It shall be the duty of the mayor to see that the city officers and departments faithfully perform their duties; to maintain peace and good order within the city; to take care that the laws of the state and the ordinances of the common council are executed and enforced within the city; to communicate by written message to the common council at least once a year a statement of the finances and general conditions of the affairs of the city, and with such recommendations in relation thereto as he may deem proper; to give such information in relation to the same as the common council may from time to time require; and to call a special meeting of the common council whenever in his judgment it is required by public necessity. He shall also receive and examine into all complaints made against any city officer for neglect of duty or malfeasance in office.
N.Y. Second Class Cities Law § 54