Pizza, Hot Dogs, Falafel, Burritos, Hamburgers and Gyro's are everywhere in New York. I'm going to say those are food specialties. Go for Pizza from small pizzaria's. Hot dogs from Gray's Papaya. Burgers from Diners, like Cozy Burger. Burrito's from real Mexican joints and Gyros and Falafel, well that's everywhere, in stands and in small restaurants.
For Donuts, another specialty, go to Dunkin'Donuts or KrispyCreme.
If you want o go even breader, tastier and more refined, I would eat the "specialties" by neighborhood. Chinatown, Little Italy, Ave B for Indian, etc.
There are so many things New York is known for!
New York Cheesecake for one!
and that's just regarding food specialties!