The most common and most likely answer to this crossword clue is the 2 plus 5 plus 4 plus 4 letter word ON THEIR LAST LEGS
I've seen this crossword clue in The Times
O
N
T
H
E
I
R
L
A
S
T
L
E
G
S
Buddy explains!
The straight part
We know that "close to defeat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case close to defeat = on their last legs.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.