Leave a pet in custody of vet endlessly (6)
I believe the answer is:
vacate
'leave' is the definition.
(to vacate a place is to leave it)
'a pet in custody of vet endlessly' is the wordplay.
'pet' becomes 'cat' (cats are often kept as pets).
'in custody of' indicates putting letters inside (inserted letters are taken into custody by others).
'endlessly' means to remove the last letter.
'a'+'cat'='acat'
'vet' with its final letter taken away is 've'.
'acat' going inside 've' is 'VACATE'.
(Other definitions for vacate that I've seen before include "Depart from, leave empty" , "Move out and leave empty" , "Leave (a position)" , "Withdraw as from premises" , "Quit" .)