The US doesn't really have that class of store for some reason. Even in urban areas, there tend to be either full-fledged grocery stores or "convenience stores" like 7-11 which have minimal fresh food. There are certainly exceptions but they tend to be local one-off markets. AFAIK, there aren't any widespread versions of the express/metro markets like you see in the UK.

There are sort of tiers in the U.S. I would think of something like this for things we have in the Southeast US:

* Convenience Store

* Dollar store (a little more food options at places like Dollar General/Tree/Family Dollar, usually with a small refrigerator section and eggs and milk and juice)

* "Mini" grocery stores, like Piggly Wiggly Express or Dollar General Market

* Stores with mostly food and limited selection like Aldi

* Supermarkets like Kroger and Publix

* Supercenters like Super Walmart and Super Target with full supermarket selection and lots of other stuff