Teacher's pet almost (5)
I believe the answer is:
rabbi
'teacher's' is the definition.
(Jewish teacher)
'pet almost' is the wordplay.
'pet' becomes 'rabbit' (a rabbit might be kept as a pet).
'almost' means to remove the last letter (most of the word but not all of it).
'rabbit' with its last letter taken off is 'RABBI'.
(Other definitions for rabbi that I've seen before include "Hebrew teacher" , "Jewish minister" , "Sacks, for example" , "Leader of worshippers" , "Religious leader of Jewish congregation" .)