I live in a European city and in a 500 meter radius I probably have ten places where I can buy some sort of vegetable, fruit, dairy product, fresh bread, etc.

In my American city, in a 500 meter walking distance there's literally nothing but single-family houses, churches, and a school. It is prohibited by law for there to be anything else! (I say "walking distance" instead of "radius" because there are stores on the other side of the freeway, but if you can't get to them they don't count, right?)

What I did wonder: in poor urban areas in the US, are there no small shops catering mostly to immigrants?

Sure, if there are allowed to be (see previous paragraph). Otherwise, shops catering mostly to immigrants have to be located on the main stroad and poor immigrants have got to drive to them just like everybody else. (Or take one of the private bus services that sometimes exist to cater to those areas.)