The most common and most likely answer to this crossword clue is the 7 plus 6 letter word EXPRESS REGRET
I've seen this crossword clue in The Daily Telegraph
E
X
P
R
E
S
S
R
E
G
R
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T
More clues leading to the result EXPRESS REGRET
Buddy explains!
The straight part
We know that "Make a speedy apology?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case make a speedy apology? = express regret.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.