There are generally a few different types of homeless people.
Drug addict.
A life style choice (easier to run away and have no problems).
Those people who have Fallen on VERY hard times and slipped through the net for what ever reason.
For obvious reasons a drug addict wouldn't want to go to prison, for fear of not being able to chase the dragon.
The other two categories will sometimes try get arrested to spend the night in the cells, get a warm meal and bed for the night. The police know this tho. Take into considerations that the non smack head ones will have a moral compass so committing a big enough crime to get sent away for a while won't be on the cards, and I'd imagine being in prison long term is deffo not what they want, even if it means they get looked after.
And lastly the point of the lifestyle choice. These people are running away from something, they don't want to have to face problems/etc. There are charities to offer help, systems etc to get them some form of fixed a abode, often they won't want it though, it means being sober/being a part of society.
In general, in England being homeless is a choice. We have a lot of safety nets and you gotta really want to not be helped to end up on the streets. For what ever reason that may be.
So in short
Yes some homeless people will commit petty crimes to get off the streets for a few nights when the weather or what ever is really bad. Long term prison (I.e. Committing a big crime) is off the cards for most homeless for the reasons above.