Right to live in the city though it did have strings attached (5)
I believe the answer is:
rebec
'it did have strings attached' is the definition.
(historical type of stringed instrument)
'right to live in the city' is the wordplay.
'right' becomes 'r' (common abbreviation).
'to live' becomes 'be' (synonyms).
'in' is an insertion indicator.
'the city' becomes 'EC' (the City of London postcode).
'be' inserted within 'ec' is 'ebec'.
'r'+'ebec'='REBEC'
'though' acts as a link.
(Other definitions for rebec that I've seen before include "Mediaeval stringed instrument" , "Medieval bowed instrument" , "Old stringed instrument" .)