The most common and most likely answer to this crossword clue is the 10 letter word SACCHARINE
I've seen this crossword clue in The Daily Telegraph
S
A
C
C
H
A
R
I
N
E
Buddy explains!

The straight part
We know that "Sickly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case sickly = saccharine.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 SACCHARINE mean?
SACCHARINE noun- Sentimentalism
- Of or relating to sugar.
- Excessively sweet in action or disposition; syrupy.
- Sentimental or romantic to the point of ridiculousness.
- Of or relating to saccharin.