Increase number entering stage immediately (8)
I believe the answer is:
lengthen
'increase' is the definition.
(I know that lengthen is a more specific form of the action increase)
'number entering stage immediately' is the wordplay.
'number' becomes 'n' (n can represent any number).
'entering' is an insertion indicator.
'stage' becomes 'leg' (a leg is a stage of a competition).
'immediately' becomes 'then'.
'n' put into 'leg' is 'leng'.
'leng'+'then'='LENGTHEN'
(Other definitions for lengthen that I've seen before include "Extend or draw out" , "Make or become longer" , "increase in duration" , "Stretch" , "Protract" .)