The most common and most likely answer to this crossword clue is the 6 letter word AIRILY
I've seen this crossword clue in Financial Times
A
I
R
I
L
Y
More clues leading to the result AIRILY
Buddy explains!

The straight part
We know that "in a flippant manner" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in a flippant manner = airily.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 AIRILY mean?
AIRILY adverb- In an airy manner.
- Lightly