The most common and most likely answer to this crossword clue is the 8 letter word TRUFFLES
I've seen this crossword clue in The Guardian
T
R
U
F
F
L
E
S
Buddy explains!

The straight part
We know that "Delicacy" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case delicacy = truffles.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 TRUFFLES mean?
TRUFFLES noun- A confection having a center of ganache and an outer coating of powdered cocoa or chocolate.
- Any of various edible fungi, of the genus Tuber, that grow in the soil in southern Europe; the earthnut.