Self-important and drunk - also powerful (4,3,6)
I believe the answer is:
high and mighty
'self-important' is the definition.
(I've seen this before)
'drunk also powerful' is the wordplay.
'drunk' becomes 'high' (I've seen this before).
'also' becomes 'and' (both can mean in addition).
'powerful' becomes 'mighty' (similar in meaning).
'high'+'and'+'mighty'='HIGH-AND-MIGHTY'
'and' acts as a link.
(Other definitions for high and mighty that I've seen before include "like the Sun, perhaps? - with 9a,10a" , "Stuck up" , "Self-important" , "Arrogant - pompous" , "Domineering" .)