I am also a single mother of a 2yr old on welfare. Im 22 now but at that age I found that finding jobs in retail the easiest to get, like working at Ross or walmart even if its temporary and put open availability. If you don't put open your application won't even be looked at.
Another option would be to go to college. That's what i'm doing. My local welfare office pays for everthing from parking passes, to classes, and all of my books (new books not used). The college also offers grants. The more units you take the more money you get and all you have to do is maintain a c average but it is always better to do better because you want employers to look at your transcript or potential univerisites to see that you are a reliable person and can do your job very well instead of just getting by if you know what I mean. The grants the welfare can't touch, so all the money you receive you don't have to report it and it won't count against what they give to you.
Its still hard im not going to lie about that, but it does make things a little easier. If you're undecided about a career try doing either basic classes or take a test and it could tell you the best job for you. If you want to be a stay at home mom after college, take child care classes. My daycare receives 536 a month for watching my daughter full time (welfare pays for this also). So that's also a option. If you really think about it watching two kids full-time could possibly pay your bills, I know it will pay mine, but I chose a different career field.
Sorry for the long post, I just wanted to let you know that I am also a single mother on welfare and was homeless until recently. Now I live in a very nice apartment and go to school full time and will be graduating with three associates degrees and I just started college in the fall the year I had my daughter.
Good Luck I'm here if you need to talk.