Influence using extremely slow activity (4)
I believe the answer is:
sway
'influence' is the definition.
(I know that sway is a more specific form of influence)
'extremely slow activity' is the wordplay.
'extremely' suggests removing the centre (only the extremes of the word are used).
'activity' becomes 'ay' (I can't explain this - if you can you should believe this answer much more).
'slow' with its centre removed is 'sw'.
'sw'+'ay'='SWAY'
'using' is the link.
Can you help me to learn more?
(Other definitions for sway that I've seen before include "Move gently" , "Ways to waver in opinion or posture" , "Move from side to side; influence" , "Move back and forth" , "Power - rock" .)