The most common and most likely answer to this crossword clue is the 7 letter word CATNAPS
I've seen this crossword clue in The Times
C
A
T
N
A
P
S
More clues leading to the result CATNAPS
Buddy explains!

The straight part
We know that "gets fitful sleep" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gets fitful sleep = catnaps.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 CATNAPS mean?
CATNAPS noun- A brief, light sleep.
- To take a catnap, to take a short sleep or nap.
- To kidnap a cat.