Eastern writer with regret regularly gets to appear (6)
I believe the answer is:
emerge
'appear' is the definition.
(emerge can mean to appear or arrive)
'eastern writer with regret regularly' is the wordplay.
'eastern' becomes 'e'.
'writer' becomes 'me' (the person who wrote the clues).
'with' says to put letters next to each other.
'regularly' means one should take alternating letters (regularly take one letter, leave next etc.).
The alternating letters of 'regret' are 'rge'.
'e'+'me'+'rge'='EMERGE'
'gets to' acts as a link.
(Other definitions for emerge that I've seen before include "Suddenly appear" , "Issue" , "Emanate" , "Come into greater prominence" , "Come out or surface" .)