Some anecdata:
I spend a decent amount of time in a small (~3k) town in the American west that's the largest place for about two hours in any direction. It has one grocery store. Prices and selection in this (non-chain) store are genuinely appalling. Fresh fruit and veg is expensive and generally low in quality. As a result, nobody but the occasional out-of-towner fisherman buys it, so there's no incentive to stock more/better stuff. It's kinda like a slightly scaled up version of the basket of brown bananas in the bodega: a desultory corner of the store where there's visible dust on the apples.
What do many folks buy instead? TV dinners. Lots of them. It's not uncommon to see a cart in the checkout line that's nothing but about twenty TV dinners, cheap beer and maybe a loaf or two of wonder bread. Not everyone lives like that, but it's common enough that it's rare for me to leave that store without feeling kinda sad.
So, yeah, food deserts aren't just an urban problem.
n.b. on reflection, a partial counterexample: the Walmart in the nearest larger (~20k) town, despite having more produce and similar goodies on offer, also does a very brisk trade in TV dinners. Access to nutrition is probably relevant, but there's got to also be broader cultural/economic/wellness issues at play.