How do I change the spelling of my first name on my birth certificate in New York state

I would think the first thing to do is to go get a birth certificate with the spelling as it is.
Then, you would ask a court to let you have a hearing for such name change.
While it may be just a matter of spelling, if such spelling is of major change, then you would, in fact, be changing your name. That is a long process needing court action.
There are many things you would have to do after making such change such as changing the name on credit cards, bank accounts, insurance papers, wills... etc etc etc.

If it's a simple matter of putting a t before an f... or flipping an I... e... in place of a e... I... then why bother, just use the spelling you prefer and it will become common.
If such action is not, in any way, an attempt to circumnavigate a legal problem, then there is no reason not to just change the spelling as you see fit.

My name on my birth record is spelled genice but the correct spelling is cenice how can I change this to the correct spelling for free