I think we need to distinguish between the rough sleepers, who are highly visible every night in Manchester, and the much greater (but less visible) number of homeless families who are housed in hostels and B&Bs because they could not afford to pay their rent.
The former are mostly individuals who are on the street because of all sorts of personal problems - relationship breakdown, losing a job, drink, drugs and, yes, mental health problems as ringi says. And the reluctance of landlords to house such individuals is understandable. But their numbers are relatively small - I read somewhere that there are around 100 rough sleepers on an average night in Manchester.
But the much bigger problem, but it's a less obvious one, is the dramatic increase in homeless families, who, by and large, don't have mental health problems but simply can't pay the rent. The MEN article to which I posted a link earlier suggested that there are about 1800 children currently in temporary housing in Manchester, and that the figure is rising because of factors such as benefit cuts.
It's those families who would benefit most if rents were to come down through a significant increase in the availability of housing.