One thing I hear repeatedly online in health and nutrition forums is that it's impossible to afford a healthy diet when you're poor. Nothing could be further from the truth. A healthy diet (at least in the US) can be had for very little money. A change in eating and cooking habits is all that's required. Here are a few tips:
Only buy real food and only buy unprocessed food (or in the least processed form possible). Buy dried beans instead of canned beans. Buy pumpkins instead of pumpkin puree. Buy milk to make your own yogurt, cheese, and other dairy products. Buy bulk spices to make your own taco seasoning and other spice mixtures that usually come in expensive little packets. Make your own condiments from scratch--like mayonnaise, horseradish sauce, mustard, ketchup, and salad dressings.
Buy your grains, beans, nuts, seeds, spices, and coffee in bulk. Look into buying clubs and food co-ops--bulk foods can be super cheap bought that way.
Reduce or eliminate meat. Think of meat as a condiment. A little really does go a long way.
Grow your own food. Make it a priority to have access to a patch of land for gardening. If you don't own land, look for community gardens, go in with a friend who has land, garden in an elderly person's yard for a share of the produce, garden on your balcony or with cheap fluorescent grow lights, plant a secret garden on public land.
Plan your meals and use up all leftovers. Waste nothing. Save vegetable peels to make into vegetable stock. Save bones to make bone broth. Strain tallow and lard drippings when you cook meat and save for re-use.
Don't use coupons. Coupons are almost always for overly-marketed, over-priced items (usually processed foods). Those aren't the foods you should be buying and they aren't cheaper than bulk foods and whole natural foods.
Don't buy boxed cereals--they're a nutritional disaster and insanely overpriced. But whole grains are fine for cereal--rolled oats, wheatberries, buckwheat groats, etc. These you can buy in bulk at a natural food store or through a buying club or co-op.
Grow sprouts. Sprouting seeds are cheap and fresh sprouts are packed with nutrition.
Don't stock your cupboards with empty calories. Every calorie should be packed with nutrition. So--no chips, pretzels, cookies, pastries, refined sugar, refined flour, white rice, toaster pastries, puffy cereal, candy, soda, or fruit drinks.
If you see a good deal on produce at a roadside stand or at the grocery store, stock up. Freeze, can, dehydrate, ferment, or root-cellar the excess so it doesn't spoil.
Forage. Learn to safely identify wild edibles. Eat dandelion greens, lamb's-quarters, chickweed, etc. Pick fruit from neglected fruit trees (ask permission first if you'll be on private property). Find farmers who allow public gleaning after harvest.
Buy full-fat everything. Every calorie counts and you need fat in your diet. When you give up processed foods you eliminate a lot of unhealthy fats. Replace those fats with whole milk, whole yogurt, butter, lard, tallow, fatty fishes, virgin coconut oil, extra-virgin olive oil--all of these fats in their natural state are great for the body.
Cook and bake everything from scratch--all your soups, stews, breads, pasta, sauces, etc.
Go in with friends to purchase a side of beef or a whole hog and do the final cuts and processing yourself.
Raise backyard livestock--chickens, meat rabbits, or whatever you have space for. Make it more economical by selling excess eggs, breeding bunny pairs, etc. to cover feed and veterinary costs.
If it's economical in your state or province to do so, get a hunting or fishing license and try to bag your limit.
Make all your beverages at home--fruit and vegetable juices, gingerale, rootbeer, yogurt and kefir based drinks, beers, wines, teas, etc.--but mostly drink water.
You don't have to do all of these things to eat cheaply, but obviously the more the better. The change with the biggest pay-off is giving up processed foods, and the second biggest is probably finding a way to grow as much of your food as possible. But do what you can--there are many ways to drastically cut your food budget and still eat very, very well.