I agree it needs some thought, but it is easily doable if you're happy to eat some simple meals.
I spend max £100 per week for a family of four, normally shopping at Tesco but sometimes Lidl or Morrisons (luckily we have lots of supermarket options locally.)
Cleaning stuff.....we don't use washing up sponges, I have reusable microfibre cloths that I do all cleaning and washing up with (colour coded for kitchen, bathroom etc as I don't like the thought of washing dishes with the bathroom cleaning cloth 🤣) plus a long handled silicone scrubbing brush that I've had for years for cleaning bottles, scrubbing pans etc. Some initial outlay but once bought you don't need to repurchase.
Cleaning sprays I make my own multipurpose spray that I put in an old spray bottle. Big squirt of lemon juice into the bottle (I use those squeezy bottles of concentrated lemon juice as they're cheap), tsp of bicarbonate of soda, squirt of washing up liquid and topped up with water then given a shake. That is cleaner for kitchens, bathrooms etc. You can use white vinegar instead of lemon juice and some essential oil to mask the vinegar smell.
For hard floors I use concentrated disinfectant topped up with water to mop.
I stopped using fabric softener years ago so that is another thing I don't buy.
The above is really for environmental reasons and to cut plastic waste but it's got the added benefit of being cheap too.
Food, meals are simple. We only have meat twice a week or so and use lots of tinned pulses, frozen fruit and frozen veg.
Breakfast is normally porridge or weetabix. Sometimes toast, sometimes fruit and yoghurt. Porridge in particular is really cheap and you can add frozen fruit rather than fresh to keep the cost down.
Lunches are soup, sandwiches or baked potatoes. Piece of fruit on the side. A cheese and tomato sandwich plus an apple is very cheap, as is a baked potato with beans.
Dinners are omelettes with veg, egg and beans on toast, chickpea curry, egg and beans on toast, veggie pasta, vegetable risotto, three bean chilli etc. Frozen veg in curries, risottos etc is absolutely fine. I'll stick some frozen onions, peas, sweetcorn etc in with the pasta. Or fry mushrooms (really cheap) and stir in pasta and some cream cheese.
Snacks and puddings for DDs are fruit (cheaper ones like apples, pears, plums and satsumas) or yoghurt. I buy the big tubs of plain yoghurt and add honey. I buy a packet of biscuits once a week for sweet snacks. Their packed lunches for school are normally ham, cheese, tuna, hummus or egg sandwich or a pasta salad. Plus a piece of fruit (normally an apple, pear or banana), some vegetable sticks like sliced cucumber, pepper or carrot and a biscuit. Costs about £1.50 per portion at an absolute max.