Unless there are circumstances which are not stated in your post, you should be entitled to be housed by the local council. You should apply to them as homeless, but will have to convince them that your stay with your ex-partner is temporary & only because you have nowhere else to live.
You don't say what benefits you are now getting, but you should be entitled to income support and child tax credit; also to housing benefit (to help with the rent) & council tax benefit.
The child's father should pay maintenance, & this would be treated as income in claculating the income support, housing benefit & council tax benefit amounts, but not the child tax credit amount.
You may have difficulty getting these benefits while you are living in the same house as your ex-partner, unless you can prove that you are a separate household - separate finances, not co-habiting etc.
You may well need independent advice about all this, so go to your local CAB or whatever other free advice agency there is in your area.