The most common and most likely answer to this crossword clue is the 3 plus 5 plus 2 plus 4 letter word THE VICAR OF BRAY
I've seen this crossword clue in Financial Times
T
H
E
V
I
C
A
R
O
F
B
R
A
Y
More clues leading to the result THE VICAR OF BRAY
Buddy explains!

The straight part
We know that "Satirical song" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case satirical song = the vicar of bray.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.