Entirely welcoming a few in obsequious style (9)
I believe the answer is:
fulsomely
'in obsequious style' is the definition.
'entirely welcoming a few' is the wordplay.
'entirely' becomes 'fully' (I've seen this before).
'welcoming' means one lot of letters goes inside another (in sense of 'inviting in').
'a few' becomes 'some' (I've seen this before).
'fully' enclosing 'some' is 'FULSOMELY'.
(Other definitions for fulsomely that I've seen before include "displaying sickening obsequiousness" , "In an unctuous, insincere manner" .)