Sarah Kliff had a useful look the other day at the state of the research on food deserts ("Will Philadelphia’s experiment in eradicating ‘food deserts’ work?"). Food deserts is a lousy term. People generally (though not always) mean "healthy food desert". And those exist. They might be largely due to lack of demand. Well, lack of demand under a messed up status quo where some of the unhealthy food is very cheap, and some of the healthy food is expensive, and where sweet foods are marketed to kids, etc.
My point is, you stick a tomato in a store that didn't previously sell them, and the research so far says probably not that many people are going to buy it. If that's indeed true, the necessary policy might be something more in the direction of vastly cutting the over-subsidization of many unhealthy foods, and perhaps increasing the subsidization of some healthy foods, and putting more restrictions on advertising. Only after you did that would the corner stores find it in their own interest to provide fruits and vegetables, etc. In fact, at that point, the bodegas would change the product line to meet the demand, with perhaps no intervention necessary. Perhaps?