Hi OP,
let’s start with the basics. Do you want to continue staying in the property? If so, check if you can defend s21 (this can only be done in v limited circumstances (e.g the landlord did not protect your deposit) and often only delays repossession). Google ‘defend s21’ or speak to CAB or a free law centre.
Assuming you want to move out, I presume you applied to the council to rehouse you. There are certain ‘tests’ you need to pass in order for the council to owe you a duty to rehouse you. As a single mother you would be considered in priority need. If they can ‘prove’ that you are not separated from your ex then you may not be considered in priority need as not vulnerable enough. Even if you still live with your ex, as long as you are separated and a single parent the council should consider you a vulnerable person in priority need.
then there’s the issue of whether you’ve made yourself intentionally homeless (for example if you are being evicted for rent arrears and if the council thinks that you had plenty of money to pay your rent but didn’t budget properly).
Child maintenance does not affect your entitlement to be rehoused under homelessness law (and it doesn’t affect your entitlement to means tested benefits like UC).
Seek advice from FREE organisations (CAB, advice centres) and get 2 opinions - I’ve see a lot of poor advice being given.
Please dont believe everything you read on here as some posters, although kind hearted and wanting to help, wont know the ins and outs of homelessness law
And finally- if you are on benefits and if the council does rehouse you but puts you in private accomodation instead of council housing (this tends to happen in London as not enough social housing), make sure your rent is not higher than the Local Housing Allowance for your area.