The most common and most likely answer to this crossword clue is the 11 letter word EXASPERATED
I've seen this crossword clue in Independent
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More clues leading to the result EXASPERATED
Buddy explains!

The straight part
We know that "Annoyed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case annoyed = exasperated.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 EXASPERATED mean?
EXASPERATED verb- To tax the patience of, irk, frustrate, vex, provoke, annoy; to make angry.
- Having one's patience greatly taxed; greatly annoyed; made furious.
- Made worse or more intense.