The most common and most likely answer to this crossword clue is the 5 plus 6 letter word COUCH POTATO
I've seen this crossword clue in The Guardian
C
O
U
C
H
P
O
T
A
T
O
Buddy explains!
The straight part
We know that "Lethargic spectator" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lethargic spectator = couch potato.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.