The most common and most likely answer to this crossword clue is the 7 plus 3 letter word PLASTIC BAG
I've seen this crossword clue in The Daily Telegraph
P
L
A
S
T
I
C
B
A
G
Buddy explains!
The straight part
We know that "former supermarket giveaway" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case former supermarket giveaway = plastic bag.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.