About the idea of "city-owned grocery stores" from Mamdani, it’s well to understand that the reality of food deserts in poor neighborhoods,
raising the cost of being poor, is well documented. Food deserts are just another failure of unregulated capitalism.
The problem is real and his proposal is at least a step toward addressing it.

Is it, though? A step towards addressing it?

The research on the effects of introducing a new supermarket into a food desert don't really support the idea that it will change any of the things that presumably it's targeted towards addressing. People in the area generally still buy the same types of foods that they did before. IOW, the "food desert" is a symptom of people's food purchasing habits in an area, not the cause of it - supermarkets can't survive in those areas because their customers won't buy the mix of products that a supermarket provides, not the other way around.

That's the real problem with this kind of proposal. These "food deserts" are usually fairly well supplied with certain kinds of foods - after all, food is a necessity, and even unregulated capitalism is pretty good at supplying things people have to buy. It's just that they're the sort of foods you find in convenience stores, dollar stores, and the like. These are foods that are unhealthy if they form the entire basis of a diet, which is why health experts discount them and wish that low-income people would eat a lower proportion of them. But....they are non-perishable, cheap, require little to no time or equipment or other ingredients to prepare and make tasty. Which is part of why they end up making up a larger proportion of food purchases of lower-income folks, who can ill afford spoilage, often have limited access to cooking equipment or methods to make other kinds of foods easily, have limited free time and/or social capital for cooking, have smaller housing (and thus less space for cooking and storing food, especially perishable food) and are oft-times buying the cheapest calories with much of their food budget.

Which is why a grocery store is probably not a great vector for trying to change this dynamic. It's a very visual thing to do - a really big thing that someone can stand in front of at a ribbon-cutting - but far less effective at actually making changes than trying to address the factors described above, which have almost nothing to do with what food is being sold in the neighborhood.