Briefly browse city (4)
I believe the answer is:
graz
'city' is the definition.
(I have seen 'Austrian city' mean 'graz' so perhaps 'city' could also mean 'graz')
'briefly browse' is the wordplay.
'briefly' means to remove the last letter.
'browse' becomes 'graze' (synonyms).
'graze' with its final letter taken away is 'GRAZ'.
(Other definitions for graz that I've seen before include "European capital" , "Austrian location" , "somewhere in Austria" , "Austrian city" .)