Coming right after a competitor (7)
I believe the answer is:
arrival
'coming' is the definition.
(I know that coming can be written as arrival)
'right after a competitor' is the wordplay.
'right' becomes 'r' (common abbreviation).
'after' is a charade indicator (letters next to each other).
'competitor' becomes 'rival' (synonyms).
'r' after 'a' is 'ar'.
'ar'+'rival'='ARRIVAL'
(Other definitions for arrival that I've seen before include "Coming - comer" , "Appearance (at a place)" , "The A in ETA" , "Advent" , "Attainment of destination" .)