I've heard this argument a lot, but personally I've never been to an area in the US where you couldn't get to a Walmart or similar "cheap" supermarket in a reasonable drive and where you couldn't find basic fruits/vegetables.
Sure, it might not be your fancy organic bodega but they've got everything you need to eat right and it's not that expensive.
I mean, hell, even without fresh produce, you can buy rice, canned beans (black, pinto, garbanzo) and (non-syrup) canned fruit and that will be some of the cheapest food in the store.
It's really not that hard to eat healthy for cheap if you know what to buy.
(Edit: perhaps misunderstood what you were referring to, but if a food desert is say 10 miles without access to fresh groceries, not having a car makes that much more difficult. Even with a car, that means a larger time commitment. So I see that as part of the reasoning of what a food desert is and why that is a barrier to access to healthy food.)
http://www.nytimes.com/2012/04/18/health/research/pairing-of...
Anecdotally this makes sense to me. I see lots of parents around me-- poor and moneyed, white and of-color-- teaching their kids to eat shit. The Standard American Diet is a plague on the entire nation, across all demographic lines.
TFA is about racial inequalities in treatment of diabetes and other chronic health problems. There's no need to further adorn this crisis with contrived issues.