Put on important match between local teams - event at fete, maybe (6,5)
I believe the answer is:
donkey derby
'event at fete maybe' is the definition.
'put on important match between local teams' is the wordplay.
'put on' becomes 'don' ('don' can be a synonym of 'put on').
'important' becomes 'key' ('key' can be similar in meaning to 'important').
'match between local teams' becomes 'derby' (I've seen this before).
'don'+'key'+'derby'='DONKEY DERBY'
(Other definitions for donkey derby that I've seen before include "fair contest" , "Asinine event" , "Form of race" , "Riding race" , "Race between competitors on asses" .)