The most common and most likely answer to this crossword clue is the 6 plus 2 plus 1 plus 4 letter word BEATEN BY A HEAD
I've seen this crossword clue in The Daily Telegraph
B
E
A
T
E
N
B
Y
A
H
E
A
D
More clues leading to the result BEATEN BY A HEAD
Buddy explains!

The straight part
We know that "Narrowly failed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case narrowly failed = beaten by a head.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.