Influence with opening tweet (5)
I believe the answer is:
swing
'influence' is the definition.
(I know that influence is a is a more specific form of the action of swing)
'with opening tweet' is the wordplay.
'with' becomes 'w' (short for with).
'opening' indicates putting letters inside (some letters open a gap in others).
'tweet' becomes 'sing' (birdsong may be tweeted).
'w' put within 'sing' is 'SWING'.
(Other definitions for swing that I've seen before include "Jazz-like dance music" , "Big band jazz" , "Golfer's movement" , "Movement of a golf club" , "Change opinion" .)