The most common and most likely answer to this crossword clue is the 11 plus 7 letter word POLITICALLY CORRECT
I've seen this crossword clue in The Times
P
O
L
I
T
I
C
A
L
L
Y
C
O
R
R
E
C
T
Buddy explains!
The straight part
We know that "Avoiding offence, in a way" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case avoiding offence, in a way = politically correct.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.