I feel incredibly sorry for her children, who are already suffering. I don't think she wants a life on benefits. She got pregnant when she was young and stupid, and is struggling to cope with her day to day life now. I think the effect of the benefit cap will be to reduce her family to destitution, and this will damage her children further - something for which we will all foot the bill for in the future.
As a child brought up in poverty, I would say without doubt it has been the number one driver in getting me into a position that most people would consider reasonably wealthy. Not "a yacht in Monte Carlo and a private jet" wealthy, far from it, but a family home in London with no mortgage, a range of investments, good pension to come, more money coming in each month that I need to spend, no money worries wealthy.
I vowed to myself from about the age of 7 or 8 that I would not be poor as an adult. I knuckled down at school, a deprived primary, and got into a good secondary. Did well, good job, worked hard, moved in different circles to my parents, married a shrewd cookie, made some good investments in property and business, and so on.
So we may not end up footing the bill in the future for all of these deprived kids. Everyone gets to make choices.
The state must provide a safety net. A real safety net stops you from falling to your death when you fall of the tightrope. But it isn't comfortable, it's made of thin netting and because it's not a nice place to be, you want to get off it asap. It is temporary. The welfare safety net should be no more. It should provide the bare minimum needed. No one in this country should starve, be homeless, or not have access to free education and healthcare. But people who make bad choices should be (relatively) poor, and their children, whilst clothed, fed and watered, unfortunately have to grow up in poverty.
Any other way just makes people content to accept the safety net as a permanent means of support, and in the long run will be bad for children, and with more children in that position.