Tetari, a classic NY sauce is not chunky and is not cooked before it goes on the pizza. Depending on the place it ranges from puree to crushed, and some of the higher end pizzerias crush their own whole canned
tomatoes and thicken with paste. There is a wide array of tomato viscosities available to pizzerias and many start with a very thick paste and add water to get it to their desired consistency. This is of course the most cost effective method. 85% of NY pizzerias seem to be using California grown tomatoes that are typically pretty sweet when compared to Italian tomatoes. Stanislaus is the brand I see being used the most, but there a number of others that I see as well including Old California, Escalon, or a vast array of "house brands" sold by the various distributers. A typical spice mix would contain dry garlic, oregano, salt, and maybe a touch of sugar. Of course everyone adds or takes away to find their own niche. It is not uncommon to see oil, black pepper, white pepper, basil, or onion powder in there as well. The higher end places tend to use a very simple sauce of just tomatoes and salt staying true to their neapolitan roots.