Fire people in competition (4,4)
I believe the answer is:
sack race
'competition' is the definition.
(sack race is a kind of competition)
'fire people' is the wordplay.
'fire' becomes 'sack' (dismiss someone from their job).
'people' becomes 'race' (people can mean a national group or race).
'sack'+'race'='SACK-RACE'
'in' is the link.
(Other definitions for sack race that I've seen before include "Novelty competition on school sports day" , "Jumping contest" , "athletics contest" , "event - with 10d" , "sports day event" .)