we went to the supermarket and then had a Chinese.I suppose we could put this with the count/mass differences I discussed last week. In AmE you could have Chinese for dinner or have Chinese food, but have a Chinese sounds a little like cannibalism.
This have a [insert cuisine here] construction is used for take-away (BrE; AmE = take-out) meals, rather than fine dining experiences. Other examples:
[on the great nightlife for yoof (BrE slang) in Doncaster:] ...all we can do is go into town on a Friday night. Or maybe go to the cinema and have a McDonalds. (bbc.co.uk)
When in Spain, do as the locals do...have an Indian. (Benidorm Spotlight)
When I have a Burger King I have a diet coke to offset the damage. (What Mountain Bike Forum)
In AmE, you could go to McDonalds or eat at McDonalds or have a Big Mac, but you couldn't have a McDonalds. Unless you were a franchisee, of course.
Better Half points out that in AmE you can get your coffee in a to-go cup, but in the UK it has to be a take-away cup, which might be made of paper or polystyrene (AmE=styrofoam).