Drunk had rum from this city (6)
I believe the answer is:
durham
'this city' is the definition.
(I have seen 'Cathedral city ' mean 'durham' so perhaps 'city' could also mean 'durham')
'drunk had rum' is the wordplay.
'drunk' indicates an anagram.
'had'+'rum'='hadrum'
'hadrum' is an anagram of 'DURHAM'.
'from' is the link.
(Other definitions for durham that I've seen before include "Cathedral city of north-east England" , "NE city" , "Northern cathedral city" , "Bede's resting place" , "Northern diocese" .)