Hi,
I’m a single mum (also 34) working part time and I own my own house there is hope!
Firstly if he does want to sell the house you will be entitled to half of whatever is left from sale / estate agent fees. Put this aside as a deposit for your next place in a separate bank account so that it cannot be taken into account when you apply for universal credit.
Contact CMS and get maintence made official (mortgage lenders will take maintence payments into their affordability criteria if they are officially done via CMS), then you also have your wages from part time work and universal credit payments (a lot of lenders also take UC into account now too). I went with the broker London and country who know who best to go to to get a mortgage from when there’s things like maintence and UC involved.
You will be entitled to 85% of your childcare costs back. So if you need to put your LO into after-school club or nursery to enable you to work. Then submit your receipts online and they will pay 85% back on your next payment.
If you have any questions feel free to ask as I’ve summarised all of this xx