Playwright's expression of annoyance, not quiet (4)
I believe the answer is:
shaw
'playwright's' is the definition.
(George Bernard Shaw)
'expression of annoyance not quiet' is the wordplay.
'expression of annoyance' becomes 'pshaw' (pshaw is an old-fashioned expression of annoyance).
'not' indicates named letters should be taken away.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'pshaw' with 'p' taken away is 'SHAW'.
(Other definitions for shaw that I've seen before include "Pygmalion playwright" , "Thicket; playwright" , "Man and Superman playwright" , "George Bernard -, Irish dramatist" , "Irish playwright, d. 1950" .)