The most common and most likely answer to this crossword clue is the 13 letter word INATTENTIVELY
I've seen this crossword clue in The Times
I
N
A
T
T
E
N
T
I
V
E
L
Y
Buddy explains!
The straight part
We know that "in a negligent manner" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in a negligent manner = inattentively.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.