Playwright having to wash up (4)
I believe the answer is:
shaw
'playwright' is the definition.
(George Bernard Shaw)
'to wash up' is the wordplay.
'up' indicates an anagram (letters get chucked up).
'wash' with letters rearranged gives 'SHAW'.
'having' acts as a link.
(Other definitions for shaw that I've seen before include "G. B. ---; T. E. ---" , "Irish playwright, d. 1950" , "Playwright; small wood" , "Irish author" , "making an appearance in Craiglockhart?" .)