Formally request private to be new recruit? (8)
I believe the answer is:
beginner
'be new recruit?' is the definition.
(a new recruit has just begun)
'formally request private' is the wordplay.
'formally request' becomes 'beg'.
'private' becomes 'inner' (as in one's private or inner thoughts).
'beg'+'inner'='BEGINNER'
'to' acts as a link.
(Other definitions for beginner that I've seen before include "Tyro, novice" , "Tyro, learner" , "One just starting out" , "inexperienced person" , "Fresher" .)