Degenerate Republican's falling asleep (7)
I believe the answer is:
relapse
'degenerate' is the definition.
(to relapse is to degenerate or regress)
'republican's falling asleep' is the wordplay.
'republican' becomes 'r' (political abbreviation).
'falling' is an anagram indicator (letters falling around).
'asleep' with letters rearranged gives 'elapse'.
'r'+'elapse'='RELAPSE'
(Other definitions for relapse that I've seen before include "A return to illness" , "Regress, fall back" , "Fall back (into ill health)" , "Downturn in health (after improvement)" , "Fall back like leapers" .)