A competition to fire people (4,4)
I believe the answer is:
sack race
'a competition' is the definition.
(I know that sack race is a type of race)
'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'
'to' is the link.
(Other definitions for sack race that I've seen before include "Novelty competition on school sports day" , "event - with 10d" , "'Novelty competition where entrants run with their feet in a bag (4,4)'" , "sports day event" , "Jumping contest" .)