The most common and most likely answer to this crossword clue is the 8 plus 7 letter word VENEREAL DISEASE
I've seen this crossword clue in The Guardian
V
E
N
E
R
E
A
L
D
I
S
E
A
S
E
More clues leading to the result VENEREAL DISEASE
Buddy explains!
The straight part
We know that "“A Lover’s Complaint” /“A Lover’s Complaint”" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case “a lover’s complaint” /“a lover’s complaint” = venereal disease.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.