Risqué nature of playwright’s succeeded (8)
I believe the answer is:
raciness
'risque nature' is the definition.
(being risque or racy)
'playwright's succeeded' is the wordplay.
'playwright's' becomes 'Racine's' (Racine is an example).
'succeeded' becomes 's' (genealogical abbreviation).
'racines'+'s'='RACINESS'
'of' acts as a link.
(Other definitions for raciness that I've seen before include "Ribaldry" , "Indecency" .)