Heavy drinker is rich (4)
I believe the answer is:
lush
This clue is a double definition.
'heavy drinker' is the first definition.
(I've seen this in another clue)
'rich' is the second definition.
(similar in meaning)
'is' is the link.
(Other definitions for lush that I've seen before include "Slang drunkard like rich growth" , "one wants a lot to drink" , "Lavish - boozer" , "Rich and juicy" , "Luxuriant (vegetation)" .)