Most people there get food from a bodega type place.
That being said, many are quick to dismiss them since they don't have the nicest/largest stores, or the most advanced systems. By doing so your ignoring reality, as that is how most people get their food, hence why they (generally) accept EBT.
In an urban market, these type of stores are 15,000-30,000 sqft. We have a few markets like this in my city, but they are in nicer areas.
Poor folk have corner stores that are <3000 sqft. We don't have the Asian produce markets or Mexican markets that you are thinking of. They generally don't sell produce, and if they do they low quality and 2x expensive.
We had a crisis about a decade ago because 80% of these stores were temporarily shut down after the owners were arrested for accepting food stamps for heroin. They aren't nice places that anyone chooses to do business with.
This is exactly why attempts to tackle this supposed "food desert" problem have fallen flat on their face. These areas aren't food deserts, they just don't have a mega-store every other block, thus they are incorrectly classified as a food desert.
With some basic cleanup, the USDA could focus on areas that are actual food deserts, where there is a need and their efforts won't go to waste.
Edit: removed extra comma.