The most common and most likely answer to this crossword clue is the 5 plus 3 letter word STRAY DOG
I've seen this crossword clue in Independent
S
T
R
A
Y
D
O
G
More clues leading to the result STRAY DOG
Buddy explains!
The straight part
We know that "Lost pet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lost pet = stray dog.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.