Individual finally asleep with horrible snore (6)
I believe the answer is:
person
'individual' is the definition.
(an individual person)
'finally asleep with horrible snore' is the wordplay.
'finally' says to take the final letters.
'with' says to put letters next to each other.
'horrible' indicates anagramming the letters.
The last letter of 'asleep' is 'p'.
'snore' anagrammed gives 'erson'.
'p'+'erson'='PERSON'
(Other definitions for person that I've seen before include "A man or woman" , "Bod" , "character" , "present appearing in this" , "Type" .)