Contemptuous expression puts playwright under pressure (5)
I believe the answer is:
pshaw
'contemptuous expression' is the definition.
(pshaw is an old-fashioned expression of annoyance)
'playwright under pressure' is the wordplay.
'playwright' becomes 'shaw' (George Bernard Shaw).
'under' is a charade indicator (letters next to each other).
'pressure' becomes 'p' (abbreviation).
'shaw' after 'p' is 'PSHAW'.
'puts' acts as a link.
(Other definitions for pshaw that I've seen before include "Contemptuous expression" , "Contemptuous cry" , "Voice impatience" , "Exclamation of contempt" , "scornful reaction" .)