Time to accept running award (6)
I believe the answer is:
honour
'award' is the definition.
('honour' can be a synonym of 'award')
'time to accept running' is the wordplay.
'time' becomes 'hour' (hour is a kind of time**).
'to accept' indicates putting letters inside.
'running' becomes 'on' (I've seen this in other clues).
'hour' enclosing 'on' is 'HONOUR'.
(Other definitions for honour that I've seen before include "Grace (with a presence)" , "High respect" , "Reputation" , "Great respect, perhaps among thieves" , "Respect or venerate" .)