The most common and most likely answer to this crossword clue is the 6 letter word DESIRE
I've seen this crossword clue in The Guardian
D
E
S
I
R
E
More clues leading to the result DESIRE
- Lust is urgent outside English society
- Hot feeling? Gentleman plunges into river
- Want some grades – I revised
- Queen once said to be upset by this request
- Aspiration of Parisian father
- Limits of destitute father's aspiration
- Very much want European father to support daughter
- Passion of French teacher, English!
- Fancy gentleman wearing flower
- Drugs blocking awful feeling of lover
Buddy explains!

The straight part
We know that "Lust" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lust = desire.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 DESIRE mean?
DESIRE noun- Someone or something wished for.
- Strong attraction, particularly romantic or sexual.
- The feeling of desiring; an eager longing for something.
- Motivation.
- To want; to wish for earnestly.
- To put a request to (someone); to entreat.
- To want emotionally or sexually.
- To express a wish for; to entreat; to request.
- To require; to demand; to claim.
- To miss; to regret.