Beseeches when sent off, concealing a tear drop (8)
I believe the answer is:
entreats
'beseeches' is the definition.
('entreat' can be a synonym of 'beseech')
I can't explain the remainder of the clue.
Can you help me to learn more?
(Other definitions for entreats that I've seen before include "Begs earnestly" , "Implores" , "Requests earnestly" , "Beseeches, pleads" .)