The most common and most likely answer to this crossword clue is the 14 letter word APOLOGETICALLY
I've seen this crossword clue in The Sunday Telegraph
A
P
O
L
O
G
E
T
I
C
A
L
L
Y
Buddy explains!
The straight part
We know that "With regret" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with regret = apologetically.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 APOLOGETICALLY mean?
APOLOGETICALLY adverb- In an apologetic manner; by way of defense, apology or excuse.