If it's "way too late" to house them, I guess you just have to kill them.
They kill themselves at an alarming rate. I'd say give fixing the root issue a try and see what happens. I imagine the existing homeless population wont be an issue for very long after you stop generating new homeless.
Here it's typically not the police but other city government organizations, literally hauling away containers of garbage and restoring the space. Of course it's a game of whack a mole.