The most common and most likely answer to this crossword clue is the 4 letter word AIRY
I've seen this crossword clue in The Times
A
I
R
Y
More clues leading to the result AIRY
- Farm building lacking roof is well ventilated
- Casual woolly, back of which has faded
- Insouciant Cockney's dangerous
- Enjoying the atmosphere? That’s risky in part of London
- Jaunty foxtrot rejected by Fay
- Light railway next to main road
- Magical being, not fine and delicate
- Two elves, only one female, are impractically idealistic
- Cheerful daughter avoiding butter, say
- Fresh milk etc? Not initially
Buddy explains!

The straight part
We know that "delicate" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case delicate = airy.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 AIRY mean?
AIRY adjective- Consisting of air.
- Relating or belonging to air; high in air; aerial.
- Open to a free current of air; exposed to the air; breezy.
- Resembling air; thin; unsubstantial; not material; airlike.
- Relating to the spirit or soul; delicate; graceful.
- Not based on reality; having no solid foundation
- Light-hearted; vivacious
- Having an affected manner; being in the habit of putting on airs; affectedly grand.
- Having the light and aerial tints true to nature.