According to iDriveSafely, in NYC, the city authorities would have to know the existence of the pothole 15 days in advance of your mishap, in order to reimburse you.
The New York City Department of Transportation has to know of the pothole’s existence at least 15 days before you had your accident. That’s right – if the city didn’t know, in writing, that a pothole existed, they will not reimburse you.
And
Drivers whose vehicles sustained pothole damage must file a claim within 90 days of the incident. You can file a property damage claim with the city either manually or electronically via the eClaim system used by the New York City Comptroller’s office.
They aren't citing any code details though. But they do have these (rather confusing) stats:
New York City paid out $138 million over the past several years to settle pothole claims, so some drivers are reimbursed. [...] From 2009 to 2015, drivers filed 12,286 pothole claims against the city, of which 1,549 were settled. The city reimbursed those drivers a total of $1.5 million.
As for the rest of the state:
If your vehicle was damaged on a New York state road, you’re probably out of luck. That’s because the New York State Department of Transportation will not reimburse drivers who experience a close encounter with a pothole during so-called “pothole season.” That runs from Nov. 15 to May 1, when cold weather makes potholes far more likely. Even if your pothole interaction occurred out of season, New York state limits the amount of damage payable to just $5,000.
For the latter, they do provide a law citation--HAY § 58.