Benefits work very well as long as there is no prior debt - that is, if you assume everything was peachy and sweet financially BEFORE the police had to come and take Daddy away...
However what often happens is that 'Daddy' had signed the family up to it's financial limit long before he was made to leave, so you then get single mothers locked into contracts they can't afford to maintain .... like my blardy 18 month Sky agreement that they WON'T cancel, despite pleading.
I am lucky, I had a reasonably tight rein on the finances - as tight as it could be - but as I found out when he left, we owed on the phone bill, the rent, the Sky (which I didn't even want) and he had lied to me about it all.
Neverthless, despite it being his fault, it fell onto my shoulders.
I was lucky. I didn't already have a smoking addiction to fight with (because God knows if I still smoked I wouldn't have been able to quit!) and I didn't have very much debt, and I was able to stay in our rented house.
I think it is the children of abused women who are most at risk of child poverty. It's abused women who need to start again from absolutely nothing - and it's their children who suffer the consequences.