Were you married? You refer to him as exp rather than exh so I am asusming not. If the house was bought in both names you might still have a claim to more than half as you need a home for your child. However whether you can afford to keep running that place on one part time salary is anotehr question.
Big decisions to make and a very difficult time emotionally to be making them.
Go to your doctor and get something to help you sleep. I put this off for years, but finally did it a few months ago and it helped me so much to get back on an even keel and simply getting enough sleep helped me deal with a lot of the stress in my life and to be able to focus more clearly. The doctor gave me 4 weeks of sleeping pills, i only used 10 days of them as I was worried about getting addicted to them. But i wish now i'd done it years ago.
Speak to the CAB about what you may be entitled too re the house, and what they suggest you do re the mortgage being up for renewal etc.
Phone the tax credit helpline, or put in a claim online. If you are working 16 hours a week you will get WTC and CTC and possibly other benefits like reduced council tax and help with childcare. Do not be afraid to use childcare to take a break as well as when you're working. I sometimes leave my son at holiday club for a couple of hours longer rather than drag him round the supermarket with me which stresses us both out and takes twice as long.
Ask your ex what he intends to do about providing for his child. Point him at the CSA website and suggest he uses their calculator and you come to an agreement between you now. If he starts umming and ahhing about it and not giving you anything put in a claim with the CSA right away as they only backdate calculations from when they receive your application. Though if you can get an agreement between you and he sets up a standing order directly too you it's normally better than using the CSA. But clearly only if he pays what he should when he should.
Call on friends, family, neighbours for help. You'd do the same for them after all, so don't be afraid to ask for help now.
And don't be afraid of your emotions. Yes you want to protect your child as much as possible. But she needs to see that it's ok to have feelings and be upset. Don't bad mouth her dad in front of her, but do let her know you are sad right now and that you like cuddles to make you feel better and that you love her very much.
Try and take some positive action rather than just letting everything go on around you because it's too hard to deal with. It is very hard to deal with, but you have a beautiful little girl who needs you. Think about what you need to do for her in terms of a home, food, love... and the rest will follow.
It will get easier, but not any time soon i'm afraid. Stay strong.
Gilly