I'm looking for a coming-of-age movie from the mid-to-late '90s (give
or take), set in Manhattan. Probably straight-to-video or
straight-to-cable, there were no recognizable stars (at least at the
time).

The plot, as I remember it, is about a teenage girl who runs away to
New York to experience the big city, falls on hard times, has a jerk
boyfriend who maybe cheats on her, experiences the city, etc. - but I
don't remember what the actual narrative was.

One scene I remember clearly is that she has to sleep on the street
(in an alleyway, near a dumpster) her first night in New York, and she
explains herself to a black musician who befriends and helps her out.
There are a couple of later scenes set in clubs where's he's playing
or in the crowd.

It starts in Manhattan, and features a young runaway who befriends an
older black musician and his young protoge... who go on the road to
search for the Blues...  Maybe you're thinking of Jaimie Gertz in
Crossroads, although she doesn't come into it until after the
musicians leave New York.  It's also before your stated time frame.
http://www.filmjudge.ca/c/crossroads1986.htm