Hoping not to finish pain medication (7)
I believe the answer is:
aspirin
'pain medication' is the definition.
'hoping not to finish' is the wordplay.
'hoping' becomes 'aspiring' (I've seen this before).
'not to finish' means to remove the last letter.
'aspiring' with its final letter taken away is 'ASPIRIN'.
(Other definitions for aspirin that I've seen before include "Pain-relieving drug made in Paris" , "Analgesic over in Paris" , "Common painkiller" , "Common analgesic drug" , "answer to pain?" .)