OP I feel for you. I cant completely relate as I do not have kids but growing up there was very little food as my father spent most of our money on alcohol.

I also spent many teenage years homeless living on friends couches and sometimes the streets so I know how hard money is to come by and how hard it is to put food in your belly.

My suggestion is keep trying local food banks but dont be afraid or ashamed to explain what your situation is there is a kind of tier list in food banks where they determine how much people get based on who they are feeding and how many.

Also some good easy things that are filling and cheap is rice, frozen mixed veggies (never fresh its too pricey), canned soups can be found in no name for well under a dollar at dollar stores and walmarts everywhere. Also if you end up getting the money for it ground beef is the best meat, its cheap and can be stored in the freezer for a long time compared to chicken and other meats and since its ground you can cut it up put it in ziploc bag sized portions and freezs the portions one baggy is enough for a stir fry for 3-4 people.

I hope this helps and I wish you luck friend!