The most common and most likely answer to this crossword clue is the 10 letter word ORATORICAL
I've seen this crossword clue in The Daily Telegraph
O
R
A
T
O
R
I
C
A
L
More clues leading to the result ORATORICAL
Buddy explains!

The straight part
We know that "of public speaking" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case of public speaking = oratorical.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 ORATORICAL mean?
ORATORICAL adjective- Of, or relating to oratory or an orator