One classification including artist, regularly annoying (10)
I believe the answer is:
irritating
'annoying' is the definition.
(I know that annoying can be written as irritating)
'one classification including artist regularly' is the wordplay.
'one' becomes 'i' (Roman numeral).
'classification' becomes 'rating' (rating is a kind of classification).
'including' indicates putting letters inside.
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
The alternating letters of 'artist' are 'rit'.
'rating' going around 'rit' is 'rritating'.
'i'+'rritating'='IRRITATING'
(Other definitions for irritating that I've seen before include "troublesome" , "tiresome" , "Abrasive" .)