The most common and most likely answer to this crossword clue is the 9 letter word EROTICISM
I've seen this crossword clue in Independent
E
R
O
T
I
C
I
S
M
Buddy explains!

The straight part
We know that "Amorous desire" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case amorous desire = eroticism.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.
What does EROTICISM mean?
EROTICISM noun- The state of being erotic, or of being sexually aroused
- Sexual excitement, especially if abnormally persistent