The most common and most likely answer to this crossword clue is the 13 letter word PAINSTAKINGLY
I've seen this crossword clue in The Sunday Telegraph
P
A
I
N
S
T
A
K
I
N
G
L
Y
Buddy explains!
The straight part
We know that "with great care" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with great care = painstakingly.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.
Crossword Answer definitions
PAINSTAKINGLY adverb- In a painstaking manner; very slowly and carefully.