I am a single parent, I'm currently a student and earn a small amount from self employment, so receive some housing benefit, CTC, WTC and CB.
Let's say, hypothetically, my bf moved in with me (there is no real danger of that happening for a very long time, but let's pretend). He earns £50k a year. If he moved in, as far as the system would see it, my children are his children, and therefore he would be jointly financially responsible for them. I would instantly lose all of my benefits, leaving me around £1100 a month worse off. This would leave me in a position of being no longer financially independent, and feeling like I had to go to him, cap in hand, to ask for money. Money, which often would be spent buying things for my kids.
My bf is a very nice chap and all that, but I doubt he'd cough up a grand a month to provide for me and my children. I doubt there's many blokes that would. His dd would also lose out, as through suddenly having gained two extra children, the maintenance she is entitled to would go right down.
Bearing all this in mind, I can see why many single parents are tempted to move their partner in 'on the sly'. Of course this is very risky, but only for the single parent (usually the female). As the benefits claimant it is the single parent who will be prosecuted, the partner they'd moved in would have no repercussions, even though no doubt they'd done quite well in terms of their own living costs, probably chipping in a token amount towards food and bills.
I think this makes it very hard for single parents to ever have a serous relationship, unless they happened to be a high earner themselves, so benefits weren't an issue. Or I suppose if both adults were on benefits, as they wouldn't lose out there. I'm not sure what the answer is, other than a citizens wage (which will never happen).