Heh yeah, it's a bit of a contradictory term, but it's the only 'generic' way to describe it that I know of. Most of them are branded with the store's name with a suffix like 'local', 'express' or 'metro'
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