King Edward periodically doesn't flush (6)
I believe the answer is:
redden
'flush' is the definition.
(I know that flush can be written as redden)
'king edward periodically doesn't' is the wordplay.
'king' becomes 'R' (abbreviation of Latin rex).
'edward' becomes 'Ed' (short for Edward).
'periodically' means one should take alternating letters (letters must be selected 'every so often').
The alternating letters of 'doesnt' are 'den'.
'r'+'ed'+'den'='REDDEN'
(Other definitions for redden that I've seen before include "Blush like the dawn sky" , "Reveal embarrassment in colourful way" , "Flush, become embarrassed" , "Blush, colour accordingly" , "Blush or become flushed" .)