The most common and most likely answer to this crossword clue is the 7 letter word DOGGONE
D
O
G
G
O
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E
More clues leading to the result DOGGONE
Buddy explains!

The straight part
We know that "that’s annoying" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that’s annoying = doggone.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 DOGGONE mean?
DOGGONE adjective- Damned by God.
- Used as an intensifier expressing anger.