Keen supporter’s offering, bagging victory at the outset (4)
I believe the answer is:
avid
'keen' is the definition.
(both can mean enthusiastic)
'supporter's offering bagging victory at the outset' is the wordplay.
'supporter's offering' becomes 'aid' (I am not sure about this - if you are sure you should believe this answer much more).
'bagging' indicates putting letters inside (bag can mean to obtain or contain).
'at the outset' suggests taking the first letters (the outset is the beginning).
The first letter of 'victory' is 'v'.
'aid' enclosing 'v' is 'AVID'.
Can you help me to learn more?
(Other definitions for avid that I've seen before include "In operation" , "Excessively keen" , "Zealous, enthusiastic" , "Greedily eager" , "Eager like diva" .)