This is a cryptic definition - the entire clue describes the answer in a misleading or indirect way, playing on the literal meaning of the words.
Think about what type of contest involves multiple people, where the person who starts it cannot be the same person who finishes it. The clue is describing the nature of this specific sporting event.
Answer: RELAY RACE | This is a cryptic definition describing a relay race - in this type of contest, one person starts the race but a different team member finishes it, so the starter literally won't finish.