The most common and most likely answer to this crossword clue is the 5 letter word SHAMA
I've seen this crossword clue in The Guardian
S
H
A
M
A
More clues leading to the result SHAMA
Buddy explains!

The straight part
We know that "Indian songbird" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case indian songbird = shama.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.
What does SHAMA mean?
SHAMA noun- Copsychus malabaricus (white-rumped shama), a saxicoline songbird of India, glossy black with a white rump and brown underparts, and six other species in genus Copsychus.