OK OP I'm disabled and on benefits so I did think WTF, but I can see you are genuine.
I agree with everyone else about using a spread sheet or budget sheet - write EVERYTHING down.
What debts do you have? Look at those first, if you are not paying credit cards off in full then start there. Look at balance transfers for credit cards with 0% interest rates and swap. Do not take credit cards out with you Either take cash or open another bank account with a debit card and only put £10 into it, you should not need more for day to day.
Next what does it cost you to work, both you and your partner? Do you get a coffee on the way to work? Do you buy lunch? How fast do you drive?
There are cuts to be made, if you normally call in at costa it might be worth buying a pod coffee machine and a Contigo container - yes that's an outlay but it will pay for itself.
Your shopping
Are you buying ready meals? Convenience foods? Can you buy more fresh and cook? Do you make spontaneous purchases? Stop doing that, it can work out cheaper to shop online because you are not tempted with a new cereal or a T shirt that catches your eye.
Meal plan - my plan has slots for daily meals and then I check what I have in the cupboards / freezer and only list what I need to buy.
How old are your children? Get them involved with cooking.
Do the kids have contract phones? They don't need them unless you a have a really good deal. I have unlimited calls and texts with more data than I could ever use for £3.30 a month (it's just gone up from £3) it's actually a £35 per month sim only deal but I have discounts because of being a customer for a few year.
Look at all contracts, Netflix? Sky? Phones? see if you can get a better deal.
Look at the children's activities, I noticed you said nursery so I'm assuming they are primary and below. Once a child has learned to swim do they need to continue swimming or would a family outing to the pool do to keep up?
Change light bulbs to LED ones, they save a lot of money and you can also get motion detected lights, ideal for the bathroom and hall if people forget.
Do you eat out? Have a look at where you eat and if it is good value and could you cook the same at home?
Finally join a cash back site and get in to the habit of using it for every online purchase.