I'm unsure as to the circumstances of the debt but you may be able to get a Debt Relief Order - more info on that here www.citizensadvice.org.uk/debt-and-money/debt-solutions/debt-relief-orders/
I live on a budget and I pretty much only buy clothes from charity shops. I'm lucky as in my area we have one that charges only £1.99 per item of clothing, that includes winter coats which obviously tend to be expensive.
Keep an eye on HotUkDeals - especially for cheap groceries or 'glitches' where you can often get amazing deals.
Also join the Facebook page - 10WaysToHaveMoreMoneyAsAStudentWithoutWorking - they also post some pretty good deals sometimes.
MoneySavingExpert has a huge wealth of advice regarding budgeting.
Your phone does seem to be on the expensive side. I pay £20 a month for mine, that's unlimited free phone calls, unlimited texts and 30gb of data. I'm with Three, it gets discounted because I pay Direct Debit and also because I'm a long term customer (who rings them every 12months asking for them to reduce my bill please)!!
Have a look in to Warm Home Discount. Most utility companies offer it and you're likely to be eligible, they will credit your account over winter with £140 (Can't remember if it's Gas or Electric).
When the kids are at school then layer up on clothes to reduce heating costs.
Your bills for the car seem very expensive. Do an online check to see if you'd be better off switching. Do this for your utilities too.
If you're better off with the utilities company you have then still phone them to check you are on the cheapest tarriff. Also check their website for any deals/offers - sometimes these are only available for new customers, but still phone and ask - I do! British Gas did one a few months back and I got a free £50 Amazon Gift Voucher with them.
Look at Youtube tutorials at how to cut your own hair (if you don't already). Haircuts are an unecessary cost. I haven't been to a hairdresser for 7 years and the time before that I hadn't been for another 7 years. I also cut my sons hair myself.
For days out with the kids ask in your local Facebook group if anyone knows of any free events for the kids. Take them to museums, the library, the woods etc because I imagine entrance fees for anywhere with 4 children is going to be too much.
Flog stuff you don't want! Even if you only get 50p for something pop it into a jar to save for a treat or for emergancies. Soon adds up.
Good luck OP. Show that arsehole of an ex what you're made of! 