The most common and most likely answer to this crossword clue is the 1 plus 4 plus 2 plus 3 plus 4 letter word A BIRD IN THE HAND
I've seen this crossword clue in The Guardian
A
B
I
R
D
I
N
T
H
E
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A
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D
Buddy explains!
The straight part
We know that "Tame budgie" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case tame budgie = a bird in the hand.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.