Self-important and drunk - also may yawn, initially (4,3,6)
I believe the answer is:
high and mighty
'self-important' is the definition.
(I've seen this before)
'drunk also may yawn initially' is the wordplay.
'drunk' becomes 'high' (I've seen this before).
'also' becomes 'and' (both can mean in addition).
'may' becomes 'might' (synonyms).
'initially' indicates taking the first letters.
The initial letter of 'yawn' is 'y'.
'high'+'and'+'might'+'y'='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" , "Acting as if one believes one is very important" , "Arrogant - pompous" , "Important, influential and arrogant" , "Self-important" .)