I'm posting as an adult who had issues as a kid. I actually did fine socially (as far as I remember), but somehow there were certain parents who knew that my homelife was less than ideal, and paid a little more attention to me than they did other kids. I am very grateful for these parents, because exposure to them enriched my life. However, as I grew older I became aware that I was pitied, and I took this knowledge into my adult life, feeling that any "normal" person only interacted with me because they wanted to feel like they were doing the "right" thing.
Anyway, i just wanted to share my perspective on this, as a person who has been "helped" by others. I don't really have any specific answers for you, but I wanted to remind you that this little girl's sense of dignity should be preserved in whatever you do (and so should her mother's). Giving her the impression that she is "not okay" or her family is "not okay" is only going to exacerbate the situation (how can someone feel like they have something to offer others if they are "not okay"?)
I think it's important for you to truly analyze your feelings and motivations for helping her. Don't take this the wrong way, as I'm sure you have good intentions, but it's worthwhile for you to think about whether or not you feel superior in any way to this mother, and how the little girl might pick up on that.
It's important to remember that this little girl is making choices for herself and shouldn't be viewed as a victim. Instead, it may be more helpful to focus on little ways that you can equip her to be accepted (without compromising on who she is)- teaching her social graces so to speak, while also working with the other kids on inclusiveness.
