The most common and most likely answer to this crossword clue is the 13 letter word SOFTLY-SOFTLY
I've seen this crossword clue in Financial Times
S
O
F
T
L
Y
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S
O
F
T
L
Y
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 = softly-softly.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 SOFTLY-SOFTLY mean?
SOFTLY-SOFTLY adjective- (often hyphenated) Discreet, low-key, careful.
- (western or central Africa) The potto, Perodicticus potto, a primate in the family Lorisidae.