Capital person finished after working to link both sides (8)
I believe the answer is:
londoner
'capital person' is the definition.
'finished after working to link both sides' is the wordplay.
'finished' becomes 'done' ('done' can be similar in meaning to 'finished'**).
'after' is a charade indicator (letters next to each other).
'working' becomes 'on'.
'to link' means one lot of letters goes inside another (inserted letters link the two sides of a word).
'both sides' becomes 'lr' (I've seen this in other clues).
'done' put after 'on' is 'ondone'.
'ondone' going within 'lr' is 'LONDONER'.
(Other definitions for londoner that I've seen before include "who uses capital for home?" , "One from UK's capital" , "Person from England's capital" , "Possible Cockney" , "Native of Camberwell, perhaps" .)