Sleep a smaller amount? Not quite in this city (6)
I believe the answer is:
naples
'this city' is the definition.
(Naples is an example)
'sleep a smaller amount? not quite' is the wordplay.
'sleep' becomes 'nap' (a nap is a short sleep).
'a smaller amount?' becomes 'less' (I've seen this in another clue).
'not quite' means to remove the last letter.
'less' with its final letter taken off is 'les'.
'nap'+'les'='NAPLES'
'in' is the link.
(Other definitions for naples that I've seen before include "endless sleep after being here?" , "See it and die, they say" , "City near Mt. Vesuvius" , "Major Italian city south of Rome" , "European city" .)