Good luck, and solidarity! At one point I was living on 4,000 a year, but I was single so it was much easier than what you are facing. Some of my tips may translate to family life; others may not. Mine are mostly veg-focused but not necessarily completely vegetarian as this cuts the bill dramatically, focused on buying and cooking and sometimes freezing in bulk, so some depend on freezer space and appliances. Some things work even if you don't have the appliances though:
veg box with wonky vegetables (some supermarkets do these, and there are online services too)
bulk pulses dried and soaked as the main protein in meals
big bulk cook and portion freeze of a vegetable soup with lots of pulses (eg. massive minestrone)
big batch of veg chilli with kidney beans on brown rice
if you have a blender, use a bulk bag of oats to make your own oat milk with just a blender and a sieve and funnel, takes 5 minutes, works out to 10p per litre. Substitute with some not all of dairy milk eating intake as desired (eg oatmilk on cereal, cow milk in coffee etc). You can make your own milks with other things but I've never tried as the raw ingredients (eg nuts) are way more expensive than bulk oats.
Again with a blender, making your own hummus. especially cheap if you use dried rather than tinned chickpeas. good protein, filling etc. good with crudites made from the wonky veg box.
bulk bags of cheap staples from other cuisines found in shops dedicated to those cuisines, like East Asian and Polish food shops (things like chickpeas, lentils, brown rice, bulk quinoa (as against pre-prepped pouches which are expensive and give quinoa a bad name), couscous.
Pots of herbs bought from the supermarket and kept alive for as long as possible -- much cheaper than buying sachets each time.
bulk bags of second-best apples/marked fruit -- usually they're fine just cut out the eyes.
really big tubs of yoghurt from Middle-eastern groceries -- as long as yr family eat it before it goes bad (usually lasts a while), this is great value compared to mini yoghurts.
hear me out but if you're getting any cheap old going-stale bread, you can do a breadcrumb-based pasta sauce - I was sceptical as it sounded like carbs on carbs but it turns out it's actually a thing and it's great -- with a few tinned anchovies (save the rest for sth else) and lemon, vg.
tinned tomatoes are very versatile, store well and cheap. good in a one-two-three dinner situation, where you cook, for instance, some grains dried quinoa/couscous, some greens (eg bok choy from a big bunch) and some tomatoes, and stick a poached egg on it, flavoured with miso, cheap, well balanced, delicious.
East asian leafy green veg -- often very plentiful bunches for the money, eg bok choy, and very delicious as the main thing in a stir fry, esp with bulk mushrooms rather than meat or tofu.
corn on the cob, when in season, as that is sometimes cheap and is both filling and a vegetable, tick.
Miso makes everything great -- get some really cheap bulky seasonal root veg (good time for this right now) and do a roast with just veg seasoned with miso and cheap bulk olive oil. the little sachets of miso are good as you only open what you're going to use at any one time. this makes so much food and is delicious. other way to do this is with soy sauce and olive oil. they caramelise and are very umami ish and you don't miss the roast meat.
bulk mushrooms (ie pick and weigh at the supey rather than in a punnet) instead of meat, or instead of those veg patties. much cheaper and in my opinion more delicious depending on what you do with them.
this is all quite slappably hippy, so take or leave bits as you find useful!!