The most common and most likely answer to this crossword clue is the 5 plus 5 letter word RUGBY POSTS
I've seen this crossword clue in The Guardian
R
U
G
B
Y
P
O
S
T
S
Buddy explains!
The straight part
We know that "H for “Headmaster” and “Headmistress”, but not at Harrow? (5,5)/H for “Headmaster” and “Headmistress”, but not at Harrow? (5,5)" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case h for “headmaster” and “headmistress”, but not at harrow? (5,5)/h for “headmaster” and “headmistress”, but not at harrow? (5,5) = rugby posts.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.