Annoyed, not starting to get relaxed (5)
I believe the answer is:
eased
'relaxed' is the definition.
(I've seen this before)
'annoyed not starting' is the wordplay.
'annoyed' becomes 'teased' (I've seen this in another clue).
'not starting' means to remove the first letter.
'teased' with its first letter taken away is 'EASED'.
'to get' acts as a link.
(Other definitions for eased that I've seen before include "Soothed - assuaged" , "rested" , "Helped" , "Became less painful" , "Abated" .)