Beg tree-hugging, empty eggheads (7)
I believe the answer is:
beseech
'beg' is the definition.
(I've seen this in another clue)
'tree-hugging empty eggheads' is the wordplay.
'tree' becomes 'beech' (beech tree is a kind of tree).
'hugging' is an insertion indicator (outside letters hug or embrace inside letters).
'empty' suggests removing the centre.
'eggheads' with its centre removed is 'es'.
'beech' going around 'es' is 'BESEECH'.
(Other definitions for beseech that I've seen before include "Fervently ask (someone) for something" , "request, urgently" , "Beg earnestly, plead" , "petition" , "Entreat, implore" .)