I'm certainly no expert on the Royal Hospital School, Greenwich, however looking at some newspaper articles of the time, I notice in a letter to the Times mention of Lloyd's Patriotic Fund:
"To this admirable institution [Royal Hospital School, Greenwich], which it did so much to found, the Patriotic - now Lloyd's Patriotic Fund still nominates boys; and it also, from a limited income, still makes small grants to sailors and soldiers, officers as well as men, wounded in action, and to their widows, orphans and dependent relatives ..."
Publication: The Times (London, England)
Date: Friday, May 5, 1905
I take it to mean they gave scholarships?
Also an interesting article with photos about the School from 1895:
"In 1821 the Royal Navy Asylum (founded out of Lloyd's Patriotic Fund, and containing six hundred boys) was amalgamated with Greenwich Hospital School, forming two schools - the Upper School and the Lower School - the one, however, being for children of commissioned officers and the other for children of seamen.
In 1862 the two schools were really made one, and the sons of commissioned officers were excluded ..."
As well as boarders, there were day students "under the Boreman Foundation".
Publication: Chums: An Illustrated Paper for Boys (London, England)
Author: W. J. FROST Date: Wednesday, Oct. 30, 1895