Has drinks, but no starter (4)
I believe the answer is:
owns
'has' is the definition.
('own' can be a synonym of 'have')
'drinks but no starter' is the wordplay.
'drinks' becomes 'downs' (downing is a kind of drinking).
'but no starter' means to remove the first letter.
'downs' with its initial letter taken away is 'OWNS'.
(Other definitions for owns that I've seen before include "Confesses" , "Admits possession" , "uses" , "Has possession of by right" , "Possesses" .)