When you say he is naughty at home, what kind of things are you talking about?
Before you go off to the GP, it would probably be helpful to establish if we are talking normal 6yo naughtiness or a greater level of acting up because of what he has been through or extreme behaviour that might indicate there is something more seriously wrong with him.
So what is he doing? Sometimes answering you rudely and/or failing to obey? Unable to understand your instructions at all? Being violent and destructive? Trying to set fire to the house?
I would say my ds was "normally naughty" at this age: his table manners were horrendous, he answered back at times and did not always (often) comply with demands, but was affectionate in between. Patience and a firm and consistent but good humoured approach was all he needed to gradually grow out of his undesirable behaviour.
His sister was acting out a higher level due to trauma: she went into violent tantrums where she literally didn't seem to recognise people around her but would lash out and bite and kick. She needed counselling and longterm support.
Other friends have found that the behaviour of their children has been related to various forms of SN that needed special support.
If he is just normally 6yo "naughty", then CAHMS won't be able to help him, but the GP or SureStart might still be able to point to some support for you. Don't let's forget that you have gone through a lot, without support by the sounds of it, and that struggle will have taken its toll on you. It may be that you are seeing his behaviour as worse than it is because you are exhausted and worn out with caring for other people.