The most common and most likely answer to this crossword clue is the 6 letter word RECTOR
I've seen this crossword clue in The Daily Telegraph
R
E
C
T
O
R
More clues leading to the result RECTOR
- Clergyman in park on hill
- Member of clergy’s partly indirect ordination
- Minister having somewhere to play rock
- Cleric’s right to run round City area
- Member of the clergy in park by rocky hill
- Right part of book — right for father of 6 2, for one
- Minister about to go to court with men
- Officer leaves executive head of university
- One who will say prayers in correct order
- Page on king and member of the clergy
Buddy explains!

The straight part
We know that "academy’s headmaster" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case academy’s headmaster = rector.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 RECTOR mean?
RECTOR noun- In the Anglican Church, a cleric in charge of a parish and who owns the tithes of it.
- In the Roman Catholic Church, a cleric with managerial as well as spiritual responsibility for a church or other institution.
- A priest or bishop who is in charge of a parish or in an administrative leadership position in a theological seminary or academy.
- In a Protestant church, a pastor in charge of a church with administrative and pastoral leadership combined.
- A headmaster in various educational institutions, e.g. a university.