Possibly mistress apiece in endless semester (7)
I believe the answer is:
teacher
'possibly mistress' is the definition.
'apiece in endless semester' is the wordplay.
'apiece' becomes 'each' (synonyms).
'in' is an insertion indicator.
'endless' means to remove the last letter.
'semester' becomes 'term' (I've seen this before).
'term' with its final letter taken away is 'ter'.
'each' going inside 'ter' is 'TEACHER'.
(Other definitions for teacher that I've seen before include "Coach" , "Lecturer" , "One who educates" , "Pedagogue" , "One who might give lines" .)