Go off in very old city (4)
I believe the answer is:
sour
'go off' is the definition.
(thesaurus)
'very old city' is the wordplay.
'very' becomes 'so' (both can mean 'extremely').
'old city' becomes 'ur' (ancient city state).
'so'+'ur'='SOUR'
'in' is the link.
(Other definitions for sour that I've seen before include "Acid - bitter" , "Tasting sharp like stale milk" , "Sharp- or tart-tasting" , "Acid - rancid" , "Acetic" .)