Exhausted and insensitive to rhythm (8)
I believe the answer is:
deadbeat
'exhausted' is the definition.
(I've seen this before)
'insensitive to rhythm' is the wordplay.
'insensitive' becomes 'dead' (I've seen this before).
'to rhythm' becomes 'beat' (synonyms).
'dead'+'beat'='DEADBEAT'
'and' is the link.
(Other definitions for deadbeat that I've seen before include "down-and-out" , "extremely tired" , "idle so-and-so" , "Disreputable person" , "Lazy, useless person" .)