I don't necessarily think that asking, by itself, is wrong. If someone asks I'd think there's the chance they're assume the single mother might be able to afford to lend the money. People don't always have any idea of how much money someone else has or doesn't have. Sometimes, too, someone who very much feels he'd "do the same for you" may have that kind of thinking when he asks (and assumes you don't mind if he at least asks, and also assumes you'd simply say "no" if you're able to do it).
To me, what's wrong (even if they believe they can and will pay you back) is that you're not being paid back. They may not plan not to be able to pay it back, and they may have every intention of paying you back when they can. Their mistake may well be in assuming they're getting money from somewhere (or else not having some new, unexpected, expense pop up) when they can't be 100% that all will go as planned.
Then again, though, if you lent money to people with the idea that they'd pay you back, I think you're perfectly right to draw the line (and maybe even explain to them that you can't afford to "give" them the money if you're not sure you will be paid back "within x amount of time").
Sometimes, maybe, people are just so desperate that they figure there's no harm in asking anyone close to them and anyone they think will understand "how things can be". They might just be "straight-shooters", assume you are too, and figure if you can't lend them the money you'll say "no" (and they'll understand that too). They don't necessarily have to think it's your "responsibility" to lend them money. They may just think that asking someone close for a little help is 1) what a lot of people do, 2) what a lot of people don't mind anyone else doing, and 3) something they're pushed into doing by their own dire circumstances. :/