Greek doctor doesn't finish The Tempest (4)
I believe the answer is:
gale
'tempest' is the definition.
(a tempest is a windy storm)
'greek doctor doesn't finish' is the wordplay.
'greek doctor' becomes 'Galen' (Galen was an ancient Greek physician).
'doesn't finish' means to remove the last letter.
'galen' with its last letter taken away is 'GALE'.
'the' acts as a link.
(Other definitions for gale that I've seen before include "Fierce wind" , "Strong wind" , "High wind" , "A storm of wind and rain" , "Squall" .)