How to buy groceries on a budget?

Use food bank. That is why they are available. Many churches have them

The more you cook from scratch and the less you buy pre packaged the cheaper. Find cheap sides your kids like, mine like rice and eat with dinner a few times a week.

Pinterest has tons of recipes for when you’re on a budget. I do one pot meals and I freeze left overs

I would try to apply again and follow the other suggestions.

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This also depends on where you live because I know groceries cost more in some areas of the US vs. others.
Have you tried shopping at Aldi? Buy foods that can be used multiple ways like potatoes.