For Val G., from the Not About Food board:
Our Lady of Mt. Carmel is the ethnic Polish church of Bayonne, so the midtown (20s) area where it's located is the best place to find Polish food.
There are many small Polish delis between 20th and 35th Sts. along Avenue C, which is one block west of Broadway. The one my family goes to now that Tatarowicz's is closed is Henry's on Ave. C and I think 22nd St.
Places to sit down and eat: There's a restaurant called Teddy's on Ave. C and 26th St., across the street from Shop-Rite. If you're dying for golumpkys (meat-stuffed cabbage) that's the place to go. Never been there (I prefer mom's cooking) but I hear they're dirt cheap and heavily frequented by the local Polish crowd.
Another place is Melanie's on Avenue F. . .that's 2 blocks east of Broadway, on the other side of the railroad tracks, I believe around 21st St. or so. (I'm sorry my memory is so faulty, like I said I eat Polish food at home instead of out, and I haven't been to NJ in a while.)
If ever in doubt, give them a call up at Mt. Carmel and I'm sure they can recommend plenty of places! ;-)