Almost an admission some of us exist for sport (4)
I believe the answer is:
wear
'sport' is the definition.
(sport can mean to wear or display on one's person)
'almost an admission some of us exist' is the wordplay.
I cannot quite see how this works, but
'an' could be 'a' and 'a' is found within the answer.
'us' could be 'we' and 'we' is found within the answer.
A single letter 'r' remains which might be clued in a way I don't see.
This explanation may well be incorrect...
'for' is the link.
(Other definitions for wear that I've seen before include "North-east flower" , "___ and tear" , "Tackle" , "Durham's river - erosion" , "display" .)