The most common and most likely answer to this crossword clue is the 9 letter word CAGE-BIRD
I've seen this crossword clue in The Times
C
A
G
E
-
B
I
R
D
Buddy explains!
The straight part
We know that "pet budgie, say" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case pet budgie, say = cage-bird.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.