State absorbing Western influence (4)
I believe the answer is:
sway
'influence' is the definition.
(swaying is a kind of influencing)
'state absorbing western' is the wordplay.
'state' becomes 'say' ('say' can be a synonym of 'state'**).
'absorbing' means one lot of letters goes inside another.
'western' becomes 'w'.
'say' placed around 'w' is 'SWAY'.
(Other definitions for sway that I've seen before include "Control" , "Move back and forth" , "Oscillate" , "Power - rock" , "Move from side to side; influence" .)