Beg delicacy, supporting men with no leader (7)
I believe the answer is:
entreat
'beg' is the definition.
(to entreat someone is to beg or plead)
'delicacy supporting men with no leader' is the wordplay.
'delicacy' becomes 'treat' (synonyms).
'supporting' says to put letters next to each other (in a down clue, some letters go below others).
'with no leader' means to remove the first letter.
'men' with its first letter taken away is 'en'.
'treat' after 'en' is 'ENTREAT'.
(Other definitions for entreat that I've seen before include "Ask (someone) anxiously" , "Pray" , "Beg earnestly or emotionally" , "Beg earnestly, plead with" , "Beg earnestly, beseech" .)