The most common and most likely answer to this crossword clue is the 4 plus 4 plus 6 letter word DEAR JOHN LETTER
I've seen this crossword clue in The Times
D
E
A
R
J
O
H
N
L
E
T
T
E
R
Buddy explains!
The straight part
We know that "Final, formal brushoff" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case final, formal brushoff = dear john letter.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.