| What type of housing benifit or child benifit are you talking about, What county are you from ? Were you married ( saw partner not spouse)
First you need a child custody order in place and get temporary child support order though the court against the other parent of that child.
Housing assistance, at least in the US is determined by income and financial status.
Also if you are not married to the partner where you have a joint mortage, it can be bad, since it will be hard to force them to buy your share out, and also if they do not pay even after years of you not living there, the mortage company can come after you for payment | 
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