Influential person’s large ball (3,4)
I believe the answer is:
big shot
'influential person's' is the definition.
(big shot is a kind of influential person)
'large ball' is the wordplay.
'large' becomes 'big' (synonyms).
'ball' becomes 'shot' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'big'+'shot'='BIG SHOT'
Can you help me to learn more?
(Other definitions for big shot that I've seen before include "Head honcho?" , "Important influential person" , "Important person (with a large cannon?)" , "VIP, informally" , "Informally, an important person - sounds loud" .)