Try this, it worked for me years ago. Don't know if its the same for NY as it is here in CA though. Plus with our budget it may not work at all.
I drove over a huge pothole one night that broke one of my wheels, flattened and punctured my tire, and snapped my rear passenger end link. Total damage for came out to $180 for the single wheel, $95 for the tire, and $30 for a replacement end link. These potholes were huge and damage could have been worse if I were traveling faster.
I went down to my City Hall to ask how I can file a complaint because I could have lost control and crashed with the broken wheel. Our tax dollars are supposed to go towards keeping these roads maintained too. I even included pictures of the huge pothole in daylight, the street I drove down, and the damages done to my car. After I submitted everything according to how they wanted it, the very next day, as I was driving down the same road there was already a construction crew fixing the potholes.
1 month later, I received a letter from City Hall stating that they reviewed my complaint and are reimbursing me for my wheel and tire (no end link because they determined that could happen at any time) and also added a tax fee for what I paid for them at the time. I received a check about a week later written out to me to for $297 (IIRC). I was a little upset about the end link not being payed for, but I then thought about it and figured the city still cut me a good deal.
In your case, because it was a manhole cover, I would include if there were any warning signs as well to notify drivers.
Try it. You never know. It may just work. And if you try it, let us know how it goes!