The most common and most likely answer to this crossword clue is the 8 plus 2 plus 5 letter word BACHELOR OF MUSIC
I've seen this crossword clue in The Guardian
B
A
C
H
E
L
O
R
O
F
M
U
S
I
C
Buddy explains!
The straight part
We know that "Of course, I’m learner originally trained under master of composition" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case of course, i’m learner originally trained under master of composition = bachelor of music.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.