Painting turned up after century, source of inspiration for historian (4)
I believe the answer is:
clio
'inspiration for historian' is the definition.
(Clio was the Greek muse of history)
'painting turned up after century source' is the wordplay.
'painting' becomes 'oil' (oil painting is a kind of painting).
'turned up' says the letters should be written backwards.
'after' is a charade indicator (letters next to each other).
'source' says to take the initial letters (source can mean a beginning or origin).
The initial letter of 'century' is 'c'.
'oil' backwards is 'lio'.
'lio' after 'c' is 'CLIO'.
'of' acts as a link.
(Other definitions for clio that I've seen before include "one historically inspiring" , "One of the Muses" , "In Greek and Roman myth, the Muse of history" , "one of nine sisters" , "Muse of history - car" .)