The most common and most likely answer to this crossword clue is the 4 letter word MIFF
I've seen this crossword clue in The Daily Telegraph
M
I
F
F
More clues leading to the result MIFF
Buddy explains!

The straight part
We know that "Offend" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case offend = miff.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 MIFF mean?
MIFF noun- A small argument; a quarrel.
- A state of being offended.
- (usually used in the passive) To offend slightly.
- To become slightly offended.