The most common and most likely answer to this crossword clue is the 6 letter word PINING
I've seen this crossword clue in The Guardian
P
I
N
I
N
G
More clues leading to the result PINING
Buddy explains!

The straight part
We know that "Deep desire" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case deep desire = pining.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 PINING mean?
PINING verb- To languish; to lose flesh or wear away through distress.
- To long, to yearn so much that it causes suffering.
- To grieve or mourn for.
- To inflict pain upon; to torment.
- The act of one who pines.