The most common and most likely answer to this crossword clue is the 4 plus 6 letter word SORE THROAT
I've seen this crossword clue in The Times
S
O
R
E
T
H
R
O
A
T
More clues leading to the result SORE THROAT
Buddy explains!

The straight part
We know that "complaint" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case complaint = sore throat.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.