I've worked in NYC and several southern states with people so poor they either live in their cars or in encampments by the river (illegal ones where they have to watch out for the police)
Do you know, even begging for change, you can actually avoid processed foods? it's harder, i agree, but certainly, on minimum wage, as long as you're not obsessed with organics, you can afford it.
Simple example: A friend of mine (artist friend, not at all privileged) did a backpacking trip for 6 months.
his food supplies?
50 pound bag of lentils (lugging it around gave him a free workout as well!)
He had a small portable stove (very light:>)!!)
He'd cook the lentils, add them to soup (he had a good light supply of seasonings) AND sprout them as well so as to have healthy vegetables.
You can get bulk legumes and grains online, nuts and seeds (and yes, you can order online even if you're making minimum wage and it will save you a ton off the packaging in the stores), cook a batch and freeze it and it will take you about 5-10 minutes each day to throw together all the meals for the day. Add in frozen veggies and fruit, microwaved omelettes and canned fish and you still have incredibly healthy and incredibly cheap food.