In every other Sennett reel, artist begs (8)
I believe the answer is:
entreats
'begs' is the definition.
(to entreat is to beg or ask)
'in every other sennett reel artist' is the wordplay.
'in every other' indicates alternate letters (every other letter is selected).
'sennett'+'reel'+'artist'='sennettreelartist'
The alternating letters of 'sennettreelartist' are 'ENTREATS'.
(Other definitions for entreats that I've seen before include "Implores" , "Beseeches, pleads" , "Requests earnestly" , "Begs earnestly" .)