New York City keeps its own records of birth separate from the state. If you were born there it is possible to correct your birth certificate but it isn't an easy process. The city Department of Health requires a number of specific certified documents. No substitutes accepted.
The Department of Health of NYC has proposed new rules, eliminating the surgery requirement and greatly simplifying the process.
You can find the text of the proposed change at
https://rules.cityofnewyork.us/sites/default/files/proposed_rules_pdf/p-dohmh_10-14-14_b_art._207.pdf The Department will hold a public hearing on the proposed rule. The public hearing will take place at 10 AM until 12 PM on November 17, 2014.
The hearing will be at New York City Department of Health and Mental Hygiene Gotham Center, 42-09 28th Street, 3rd Floor, Room 3-32 Long Island City, NY 11101-4132
I live on the West Coast so cannot attend myself. As I have tried repeatedly over the past 27 years to correct my name and sex this is one rule change I really want to go through.