The most common and most likely answer to this crossword clue is the 4 plus 3 plus 4 plus 4 plus 2 plus 3 plus 5 letter word WITH THE BEST WILL IN THE WORLD
I've seen this crossword clue in The Guardian
W
I
T
H
T
H
E
B
E
S
T
W
I
L
L
I
N
T
H
E
W
O
R
L
D
Buddy explains!
The straight part
We know that "despite all one’s efforts" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case despite all one’s efforts = with the best will in the world.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.