According to my Google search there are an estimated 5,600 grocery stores and 13,000 bodegas in NYC.
Exactly. It is hard to imagine there being any significant problem with food deserts in NYC. Your numbers translate to 18.6 full grocery stores per square mile and whopping 43 bodegas. That means that grocery stores are, on average, about 600 ft from each other, while bodegas are about 400 ft apart.
Thus, the only people who would have problems with access would be those with mobility issues, and Mamdani Marts do not help with that.
It seems rather far fetched to imagine 5 stores (one in each borough) providing competition on the scale to have much effect on the market unless these are gigantic big big groceries.
Five stores will also not make any measurable positive impact neither regarding access (which I have shown with your own numbers is not even an issue) nor the food budget of the average New Yorker.
So either these are pilot projects for a much more extensive rollout (at enormous cost to the taxpayers given the discounts he is promising) or else it's just a publicity stunt by Comrade Cardamom.