This person's given a subsidy, turning up Asian art (7)
I believe the answer is:
origami
'asian art' is the definition.
'this person's given a subsidy turning up' is the wordplay.
'this person's' becomes 'im' (this person means the crossword setter themselves).
'given' is a charade indicator (letters next to each other).
'subsidy' becomes 'giro' (type of subsidy given as social security).
'turning up' is a reversal indicator.
'im'+'a'+'giro'='imagiro'
'imagiro' back-to-front is 'ORIGAMI'.
(Other definitions for origami that I've seen before include "Japanese craft of paper-tearing" , "I'm a giro (anag)" , "Delicate operations" , "Art originally created in simple materials to design bird forms" , "folding stuff?" .)