Once a state prosecutor, Earl starts to mete out annoyance (12)
I believe the answer is:
exasperation
'annoyance' is the definition.
(I know that exasperation is a type of annoyance)
'once a state prosecutor earl starts to mete' is the wordplay.
'once' becomes 'ex' (prefix meaning former or onetime).
'starts to' says to take the initial letters.
'mete' becomes 'ration' (I've seen this in another clue).
The first letters of 'state prosecutor earl' is 'spe'.
'ex'+'a'+'spe'+'ration'='EXASPERATION'
'out' is the link.
(Other definitions for exasperation that I've seen before include "Annoyance" , "Disgruntlement" , "Intense irritation" , "Fury" .)