Considering this thread has included people as far back as Henry V, I don't think it has ever been reserved simply for the Duke of Sussex.
Cyril helpfully specified which Henry by using his title
Henry of Gloucester was a sickly small child. Like his older brothers he was tutored at home at first. It was his tutor who later suggested it would do him some good to be sent to school. He would go on to Eton. His older brothers went only to the naval academy and later college.His younger brother went to the same school as Henry, but later the naval route.