Live Aid's finishing in good health (5)
I believe the answer is:
dwell
'live' is the definition.
(to live/dwell somewhere)
'aid's finishing in good health' is the wordplay.
'finishing' indicates one should take the final letters (the letter which finishes the word).
'in good health' becomes 'well' (similar in meaning).
The last letter of 'aid' is 'd'.
'd'+'well'='DWELL'
(Other definitions for dwell that I've seen before include "Think moodily (on something)" , "Reside - linger" , "Settle" , "Remain" , "Inhabit" .)