Am so sorry you are having this terrible time, but I'm not sure you are going to get much help in your current situation.
I know you want your independence, but perhaps the main thing is stability for yourself and your DC at the moment? I know this probably sounds awful but you are secure where you are, and you are more likely to be able to economise and save a bit while at your mum's.
You might just have to get to grips (both you and your mum) with making the absolute best of your situation at the moment, doing what you can to get plugged into those local networks where you can find people who can help you with childcare, training and employment. That's the only way you can truly be independent.
There's no doubt it takes a lot of effort, but the one thing you really need is a secure base - which in truth you have at the moment, with minimal outgoings. No it's not your own front door, but you really do have to get to grips with what you can do with the situation you are in, however hard that is. Believe me, it would be no easier having your own place - independence brings with it the sole responsibility for everything. With no money, realistically how will you manage that?
I doubt your mum is going to chuck you and her GC out on the street anytime soon - and I doubt the housing authorities would believe that either.
Best of luck.