Self-centred, became accepted in swinging cities (9)
I believe the answer is:
egotistic
'self-centred' is the definition.
(I've seen this before)
'became accepted in swinging cities' is the wordplay.
'became' becomes 'got' ('get' can be a synonym of 'become').
'accepted in' is an insertion indicator (I've seen this in other clues).
'swinging' is an anagram indicator.
'cities' anagrammed gives 'eistic'.
'got' inserted within 'eistic' is 'EGOTISTIC'.
(Other definitions for egotistic that I've seen before include "Self-centred" , "self-absorbed" .)