Playwright: 'That's a wrap!' Not quite (4)
I believe the answer is:
shaw
'playwright' is the definition.
(Shaw is an example)
'a wrap not quite' is the wordplay.
'a wrap' becomes 'shawl' (I've seen this before).
'not quite' means to remove the last letter.
'shawl' with its final letter removed is 'SHAW'.
'that's' is the link.
(Other definitions for shaw that I've seen before include "Playwright; small wood" , "Man and Superman playwright" , "writer of plays and critical commentaries" , "Irish playwright, d. 1950" , "Thicket; playwright" .)