OP,
I still can't get to your point. What is it you want from this?
I'm one who feels that 'motherhood' is not nurtured and cherished for our lone mothers. Working hours aren't very child-friendly in the UK. I'd support flexible working for mothers whether lone or part of a couple.
However - what you appear to be saying is that you somehow connect UC to being disbanded and your life will be better.
Under UC, if you have a child under the age of 3, you're (generically) expected to seek work. That in itself is the government attempting to assist mothers to be at home with their young children. That's a choice. Some mothers need to work - some mothers don't have to. But there is a choice.
30 hours childcare are available under UC once a child is three. Yes?
So what is it you want to achieve? As many Mothers up thread have pointed out to you, they've managed to work/study with or without support.
I can't get to what you think your petition is going to achieve. I cant get to what you want.
What do you think would help you?
UC isn't going anywhere so it's a matter of getting to grips with it and making it work for you.
Perhaps if you explain what it is you want, you'll get some advice that you may find useful.