Religious leader's pet, perhaps, with no tail (5)
I believe the answer is:
rabbi
'religious leader's' is the definition.
(Jewish religious leader)
'pet perhaps with no tail' is the wordplay.
'pet perhaps' becomes 'rabbit' (a rabbit might be kept as a pet).
'with no tail' means to remove the last letter.
'rabbit' with its last letter taken away is 'RABBI'.
(Other definitions for rabbi that I've seen before include "Spiritual leader of a Jewish congregation" , "Religious leader in a Jewish community" , "endless waffle" , "Leader of worshippers" , "Religious scholar" .)