PM gets to relax after cold sore finally disappears (9)
I believe the answer is:
churchill
'pm' is the definition.
(Prime Minister Winston Churchill)
'relax after cold sore finally disappears' is the wordplay.
'relax' becomes 'chill' (I've seen this before**).
'after' is a charade indicator (letters next to each other).
'cold' becomes 'c' (eg on taps).
'sore' becomes 'hurt'.
'finally disappears' means to remove the last letter.
'c'+'hurt'='churt'
'churt' with its last letter taken off is 'chur'.
'chill' after 'chur' is 'CHURCHILL'.
'gets to' is the link.
(Other definitions for churchill that I've seen before include "PM and Nobel prize winner" , "Former statesman" , "Winston . . . . . . . . ., English PM war leader" , "He offered the British people "nothing but blood, toil, tears and sweat"" , "Old leader" .)