Talk about not finishing — throwing up contest (6)
I believe the answer is:
discus
'contest' is the definition.
(discus is a kind of contest)
'talk about not finishing' is the wordplay.
'talk about' becomes 'discuss' (to discuss something is to talk about it).
'not finishing' means to remove the last letter.
'discuss' with its final letter taken away is 'DISCUS'.
'throwing up' acts as a link.
I am not very happy about this link. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for discus that I've seen before include "Circular object thrown in athletics" , "Debate" , "flat missile" , "Field sport" , "Object thrown by athletes" .)