If at all possible, play the long game with your money. I realise that people on tight budgets don't have a lot to play with, it's expensive to be poor, because you can't bulk buy etc etc, I've been there, but if you start small, you might be able to make small changes and build up over time. Plus not everyone is on the absolute bones of their arse, it's something that can help a lot of people make their money go a little further.
For example, if you can buy a 4 pack of beans once a month, instead of buying one can every week, it can be much cheaper over time, especially if you find some on offer. Same for other canned goods, cleaning products, toiletries etc etc. Plus if you can buy 4 tins of beans this week, you don't need them the next few weeks, so have slightly more to spend on something else. If you have a little spare money the next week, you might be able to do the same with canned soup or buy a bigger bag of potatoes, etc.
So if you can shop like that, after a while you get into a rotation where you get a bigger pack less often, so you spend less overall and makes your budget go a little bit further.
My other suggestion for if you live somewhere where the cheap supermarket is in walking distance, but not close enough that you can carry shopping home, try and get a shopping trolley - there are loads on FB marketplace from a fiver.
So you can walk to the cheap supermarket and bring your shopping home in your trolley, instead of shopping at closer more expensive shop, or having to get a bus/taxi home, or getting shopping delivered from a more expensive supermarket.
Might need more than one trip a week, but I've never done a single regular weekly shop and I don't think it's the cheapest way to shop anyway.