Uttered greeting on horse? (4)
I believe the answer is:
high
'on horse?' is the definition.
(up on one's high horse)
'uttered greeting' is the wordplay.
'uttered' shows a homophone (sound like).
'greeting' becomes 'hi' (hi is a kind of greeting).
'hi' is a homophone of 'HIGH'.
(Other definitions for high that I've seen before include "Elevated; euphoric" , "Outstanding feature" , "jump" , "Western film with Gary Cooper" , "At altitude" .)