Could you cancel your TV license? I stopped watching TV years ago. There's plenty to watch on Youtube and Channel 4, Channel 5 and ITV catch up.
I would cut the sweets down. I know only pennies but then the pounds take care of themselves etc etc. Plus it would be better for your little ones' teeth. Maybe distract them with an exciting game at the park instead, on the days they would have had sweets?
Do you walk to school? If not can you? My mum couldn't drive and we had a half hour walk to school every day whatever the weather. It wasn't a problem at all - in fact I have really happy memories of those walks with my mum.
Did you say you're doing an apprenticeship? What in? How much longer do you have until you qualify?
Are there any mums near you who might be in a similar position? You could maybe do a childcare exchange so you have her child one day a week and she has yours another day, that would save you one day of nursery fees.
Some people earn pocket money renting out garage/loft space for storage. I should imagine there's demand for that in the south east so something to consider?
You say one of your main outgoings is kids clothes. Is that because they need new stuff or because you want to treat them? The youngest can mostly wear your eldest's old things right? And you can just get essentials from charity shops, no need to buy brand new except for underwear and socks. They only need 3 or 4 changes of clothes for winter and the same for summer anyway.
Do you have anything you could sell? Maybe Marie Kondo your house on Saturday and see if you have a local car boot on Sunday? You might not earn more than £20 or so but that's enough to buy food for the week, and when you've decluttered you'll feel more 'kick-ass' and ready to sort out your finances long term.
You could look at babysitting? I know that sounds really daft but if you know any trustworthy teenagers you could have them sit your little ones for £5ph and go out and babysit other children yourself for £10. Every little helps!