Bishop, say — private individual needing more experience (8)
I believe the answer is:
beginner
'individual needing more experience' is the definition.
(a beginner at something is inexperienced in it)
'bishop say private' is the wordplay.
'bishop' becomes 'B' (chess abbreviation).
'say' becomes 'eg' (short for 'exempli gratia', 'for example' in Latin).
'private' becomes 'inner' (as in one's private or inner thoughts).
'b'+'eg'+'inner'='BEGINNER'
(Other definitions for beginner that I've seen before include "Starter" , "Fresher" , "Tyro, rookie" , "Tyro, novice" , "Tiro, novice" .)