ISO Barese Italian Restaurant
I do not know of any restaurants that are specifically Barese. However, there are at least two restaurants that I am aware of in Manhattan that specialize in Pugliese cuisine, Osteria Gelsi on 9th Avenue in Hells Kitchen and I Trulli in the east 20's. Both are rather upscale places (esp. I Trulli) so it is hard to describe them as "home cooking". I have been to Osteria Gelsi and thought it was good, but nothing special. I have never been to I Trulli.
I am thinking there must be some more informal Pugliese or Barese "home cooking" type places in the outer boroughs, but I do not know any offhand.
A couple of years ago Robert Sietsema of the Village Voice did a list of the top 100 Itlalian restaurants whic highlighted a lot of the more out of the way places in the 5 boroughs. I am sure there must be some Pugliese places on his list, but I was not able to access the list on the net. I suppse you could try to email him at the Voice.