To help with budgeting in this situation?

The only bit that might be challenging is how you manage your daughters birthday.

It’s totally feasible to do a really good birthday for a 6 year old for less than £30. It’s also very easy to spend >£300 without planning to.

Planning is your friend here. As is using stuff you’ve already got in the cupboard (wrapping paper?).

I’d suggest:
Meal planning and shopping to a list. Online orders actually work well for managing a budget better for many than Lidl / aldi. Although the actual items on your list might be more, you only buy what’s on your list whereas the middle aisles can blow some people’s budgets totally.

Birthday - again, get a plan. If wrapped stuff is a lot of the excitement, then home bargains, a roll
of wrapping paper and wrapping stuff you’d be buying anyway. Shampoo, conditioner (the unicorn version not the normal boring one), tooth brush etc.

Sit down and work out the bills, take that off the total and put it aside.

Then give yourself £70 a week for groceries.

Then an amount for birthday.

Then a contingency chunk. Then you can track against that.

You've £264 a week for just food, petrol and your disposable spends? For you and 1 child? Yeah I think youll be fine.

And if you did worry you could take her to the beach, the park, the library, use coupons from the paper for cheap access to things like English Heritage. You could take snacks from home, you could play in the house, visit relatives, local museums.

That’s loads! Shop at Aldi / Lidl. Look for all the free / cheap days out in the summer hols. Lots of picnics and parks. Vinted or charity shops for presents. You’ll be fine - people manage on way less!