Arrogant greeting good worker could eventually deny (4,3,6)
I believe the answer is:
high and mighty
'arrogant' is the definition.
(I've seen this before)
'greeting good worker could eventually deny' is the wordplay.
'greeting' becomes 'hi' (hi is a kind of greeting).
'good' becomes 'g' (abbreviation).
'worker' becomes 'hand' (hand is a kind of worker**).
'could' becomes 'might' ('might' can be a synonym of 'could').
'eventually' says to take the final letters.
The last letter of 'deny' is 'y'.
'hi'+'g'+'hand'+'might'+'y'='HIGH-AND-MIGHTY'
(Other definitions for high and mighty that I've seen before include "Acting as if one believes one is very important" , "like the Sun, perhaps? - with 9a,10a" , "Arrogant - pompous" , "Domineering" , "Important, influential and arrogant" .)