Exhausted a debated possibility (4,4)
I believe the answer is:
dead beat
'exhausted' is the definition.
(I've seen this before)
'a debated possibility' is the wordplay.
'possibility' indicates an anagram (another possible arrangement of the letters).
'a'+'debated'='adebated'
'adebated' with letters rearranged gives 'DEAD-BEAT'.
(Other definitions for dead beat that I've seen before include "wanting to retire?" , "Completely exhausted (colloq.)" , "Layabout" , "Informally a down-and-out" , "An American down-and-out" .)