The competitors went after a right to be those who came (8)
I believe the answer is:
arrivals
'to be those who came' is the definition.
I can't judge whether this definition defines the answer.
'the competitors went after a right' is the wordplay.
I cannot really understand how this works, but
'competitors' could be 'rivals' ('rival' can be a synonym of 'competitor') and 'rivals' is found in the answer.
'a' is found in the answer.
'right' could be 'r' (abbreviation) and 'r' is found in the answer.
This accounts for all the letters.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for arrivals that I've seen before include "Part of airport terminal" , "first sight of JFK?" , "Section of airport" , "They reach" , "They're seen coming from a plane into the airport" .)