The most common and most likely answer to this crossword clue is the 4 plus 8 letter word HEAD TEACHERS
H
E
A
D
T
E
A
C
H
E
R
S
Buddy explains!
The straight part
We know that "Academy bosses possibly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case academy bosses possibly = head teachers.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.