The most common and most likely answer to this crossword clue is the 7 letter word ANGRIER
I've seen this crossword clue in The Daily Telegraph
A
N
G
R
I
E
R
More clues leading to the result ANGRIER
Buddy explains!

The straight part
We know that "More annoyed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case more annoyed = angrier.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 ANGRIER mean?
ANGRIER adjective- Displaying or feeling anger.
- (said about a wound or a rash) Inflamed and painful.
- (said about the elements, like the sky or the sea) Dark and stormy, menacing.