NYC Magistrates Court:
New York (N.Y.). Board of City Magistrates. First Division. (1899 - 1914). Annual report of the Board of City Magistrates of the City of New York for the year ending ..., including also the statistics for the calendar year. Freely available on HathiTrust Library - later reports are on this HathiTrust record
Annual reports of a later iteration of this court were issued from 1917 to 1963 - John Jay has only 1930,1946-1962
NYC Police Court
New York (N.Y.). Board of Police Justices. (1895- 1914?). Annual report of the Board of Police Justices of the City of New York for the year ending ... Freely available on HathiTrust Library- John Jay has only the 1876 volume - but John Jay users can access a few other volumes
see also: Paddon, M. E. (1920). The Inferior Criminal Courts of New York City. Journal of the American Institute of Criminal Law & Criminology, 11(1), 8–20. https://doi.org/10.2307/1133786
You may also request copies of historical records from the NYC Municipal Archives
Most court proceedings in NYC are under NY State or Federal jurisdiction, because they hear trials of defendants accused of breaking state and federal laws. However, historically there have been different county and city proceedings including those in Police Courts and Magistrates Courts. Some records from these courts can be found at the Municipal Archives.
For more on these historical courts please see Mary E. Paddon, Inferior Criminal Courts of New York City, 11 J. Am. Inst. Crim. L. & Criminology 8 (May 1920 to February 1921) and Cobb, W. B. (1920). A Court of Prevention: The Municipal term Court of the City of New York. Journal Of The American Institute Of Criminal Law & Criminology, 11(1), 47-59.
In present day courts that might be considered NYC courts but are really lower NY State Courts include community courts, drug treatment courts and traffic courts. As well as "Summons Court" which was featured in the NYTimes in 2012.