The simple answer is yes, it does. The complication is defining cheap, and defining healthy. It may be easier to define unhealthy “food” as something that is eaten but which provides no nutrition beyond calories. This includes anything processed, stuffed with preservatives and packaged. If it has a shelf-life of longer than a week, and contains ingredients that you can’t pronounce, let alone spell, it is probably not worth eating.
Chips, sliced bread, cereals and sodas may not cost a lot to buy, but consuming them can have very expensive health consequences. Pound for pound, you certainly pay more for real food than for manufactured food, but you are buying nutrition and health. The trick is to plan your diet so that you can include healthy foods without exceeding your food budget.
Food you grow yourself is inexpensive as are fruits and vegetables in season.Goodbread can be quite pricey, but it is not hard to make your own. Search for Ballymoe bread for an easy recipe to get you started. Staples such as rice, beans, pasta, and potatoes are cheap and are used around the world as the basis for inexpensive meals. Add whatever vegetables are cheapest at the market or on special offer at the supermarket, and a small amount of meat for flavor.
Chicken is inexpensive if you buy the whole bird or less popular parts like thighsand drumsticks in large packs. A chicken can do three meals: roasted with potatoes and vegetables, the remaining bits added to pasta or rice and the bones boiled up for stock.
Eggs are cheap dense protein, and perfectly acceptable as part of a healthy balanced diet. They are also amazingly versatile and can be prepared in dozens of delicious and filling ways.
By cutting out all the bad stuff, you should be able to afford the good.