Quickly becomes a leading polo expert (5)
I believe the answer is:
apace
'quickly' is the definition.
('apace' can be a synonym of 'quickly')
'a leading polo expert' is the wordplay.
'leading' suggests taking the first letters.
'expert' becomes 'ace' (ace is a kind of expert**).
The first letter of 'polo' is 'p'.
'a'+'p'+'ace'='APACE'
'becomes' is the link.
(Other definitions for apace that I've seen before include "flat out" , "Quick" , "(Move on) swiftly" , "at high speed" , "At a rapid rate" .)