The most common and most likely answer to this crossword clue is the 8 letter word SLIGHTED
I've seen this crossword clue in The Daily Telegraph
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Buddy explains!

The straight part
We know that "Offended" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case offended = slighted.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 SLIGHTED mean?
SLIGHTED verb- To treat as unimportant or not worthy of attention; to make light of.
- To give lesser weight or importance to.
- To treat with disdain or neglect, usually out of prejudice, hatred, or jealousy; to ignore disrespectfully.
- To act negligently or carelessly.
- (of a fortification) To render no longer defensible by full or partial demolition.
- To make even or level.
- To throw heedlessly.