The most common and most likely answer to this crossword clue is the 12 letter word EXASPERATION
I've seen this crossword clue in The Guardian
E
X
A
S
P
E
R
A
T
I
O
N
Buddy explains!
The straight part
We know that "annoyance" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case annoyance = exasperation.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EXASPERATION mean?
EXASPERATION noun- The act of exasperating or the state of being exasperated; irritation; keen or bitter anger.
- Increase of violence or malignity; aggravation; exacerbation.