Haver, beginning to abandon drug (5)
I believe the answer is:
owner
'haver' is the definition.
'beginning to abandon drug' is the wordplay.
'beginning to abandon' means to remove the first letter.
'drug' becomes 'downer' (downer is a kind of drug).
'downer' with its first letter removed is 'OWNER'.
(Other definitions for owner that I've seen before include "The person in lawful possession" , "Holder, possessor" , "Title-holder" , "possessive type" , "One who has something" .)