Time during part of the day for sport (8)
I believe the answer is:
eventing
'sport' is the definition.
(I've seen this before)
'time during part of the day' is the wordplay.
'time' becomes 't' (abbreviation).
'during' indicates putting letters inside.
'part of the day' becomes 'evening' (I've seen this before).
't' placed inside 'evening' is 'EVENTING'.
'for' acts as a link.
(Other definitions for eventing that I've seen before include "A three-phase equestrian sport" , "Equestrian competition" , "Sport consisting of dressage, cross country and show jumping" .)