Here's some info from NYC on getting the city to pay for potholes...
Getting any city, municipality or county in New York, Westchester, Long Island or New Jersey to pay for the repairs isn't easy, but following these steps will help:
File your claim within 90 days.
Make sure you have the exact address where the pothole is located.
Take pictures of the pothole and the damage.
Keep all of your repair receipts.
Unfortunately for drivers, that is not all you will need to get reimbursement from the government.
Alan Ripka, Attorney: "In order for the city to be responsible, they must have had prior written notice, or the city created the pothole, because maybe they did construction or they ripped open the street or they didn't fix the street properly."
Leon Heyward, NYC DOT: "If a driver runs over a pothole, they give a call to the comptroller's office. The controller will forward them an application and they can file a claim for damaging their vehicle.
Obviously the city can't fix potholes if they don't know about them. You can report a pothole to the city's DOT by calling 212 or 718 CALL-DOT. You can also call that number to file a complaint about damage done to your vehicle.
If your vehicle is damaged by a pothole, you can file a claim with the City of New York. Visit the New York City Comptroller's web site and use the claim form (in pdf format) or call (212) 669-3500.