The most common and most likely answer to this crossword clue is the 5 plus 5 letter word COUNT SHEEP
I've seen this crossword clue in The Guardian
C
O
U
N
T
S
H
E
E
P
More clues leading to the result COUNT SHEEP
Buddy explains!

The straight part
We know that "Try to get to sleep" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case try to get to sleep = count sheep.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.