The most common and most likely answer to this crossword clue is the 8 letter word VEXATION
I've seen this crossword clue in The Guardian
V
E
X
A
T
I
O
N
More clues leading to the result VEXATION
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 = vexation.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 VEXATION mean?
VEXATION noun- The act of annoying, vexing, or irritating.
- The state of being vexed or irritated.