Unless every fast food restaurant is outlawed and shutdown and all crap food is outlawed and removed from the shelves of every store in the U.S., you are not going to stop anyone (low income or not) from eating it.
And for those on WIC, while many crap items aren't on the WIC list, parents can still abuse it by buying things like only Cheerios for their kids to eat instead of veggies, etc. And yes, regardless of welfare, Obesity and diabetes are going to remain very common among the poor in the U.S.
In fact, the U.S. actually provides incentives for companies to use high fructose corn syrup: https://news.ycombinator.com/item?id=5795478
And for those on WIC, while many crap items aren't on the WIC list, parents can still abuse it by buying things like only Cheerios for their kids to eat instead of veggies, etc. And yes, regardless of welfare, Obesity and diabetes are going to remain very common among the poor in the U.S.