Muse of ill repute last to leave (7)
I believe the answer is:
euterpe
'muse' is the definition.
(mythical muse of music)
'ill repute last to leave' is the wordplay.
'ill' indicates an anagram.
'last to' says to take the final letters.
The final letter of 'leave' is 'e'.
'repute' is an anagram of 'euterp'.
'euterp'+'e'='EUTERPE'
'of' acts as a link.
(Other definitions for euterpe that I've seen before include "One of the Muses" , "Muse of lyric poetry and music" , "Musical Muse" , "Inspiring goddess" , "inspiring female" .)