Genuine playwright, but not British (4)
I believe the answer is:
echt
'genuine' is the definition.
(echt means genuine or authentic)
'playwright but not british' is the wordplay.
'playwright' becomes 'brecht' (Bertolt Brecht, 1898-1956).
'but not' suggests deleting specific letters.
'british' becomes 'br' (abbreviation for British).
'brecht' with 'br' taken away is 'ECHT'.
(Other definitions for echt that I've seen before include "Genuine (Ger.)" , "True" , "Really" , "Genuine -- honest" , "Genuine - authentic" .)