As silvery agrees - the one you say is your Charles Hendy at St. Vincents Industrial School, Dartford aged 13 born Chatham is the same person as the son of Charles and Catherine!
Industrial Schools like this were also reformatory schools (what today we refer to as Approved Schools)- often they were pretty bad but this one being run by the RC church appears to have been a good one. The children who would go there would be orphans, those unable to be kept at home for some reason, those who'd been in some kind of trouble, with the aim being to educate them, teach them an occupation, and discipline. Googling shows it had a very good reputation and in 1933 it officially became known as an Approved School.
Think about it logically - Hendy was an unusual name especially in Kent. What are the odds of 2 Charles Hendy's being born in Chatham on 22nd August, one 1897 and the other 1898. 'Your' Charles was at the school in Dartford in 1911, so was the other Charles yet there is only one boy of that name there. 'Your' Charles cannot be found on 1901 census, nor can the other one. We know that for whatever reason Charles William Hendy, wife Catherine, sons Charles and Francis Walter (plain Walter in 1911) are missing from 1901 census on 3rd. April - yet just one week later daughter Kathleen is born in Gillingham so they must have been around somewhere in the area (although not at the address where Kathleen was born 7 days later). Clearly, the family were in the process of moving house, with Catherine about the give birth, which could be why they were missed by the enumerator.
'Your' Charles and the son of Charles William Hendy and Catherine Varley are clearly one and the same and I really don't know how else I can convince you.
As I said earlier, my gut feeling is that the 'Horace' Varley birth in Sept. quarter 1897 (who apart from birth entry is never seen again) is the later to be 'Charles', son of Charles and Catherine, nee Varley (who didn't marry until 1899) and that the birthdate of said 'Horace' is likely to be 22nd August. As the couple weren't yet married he would have been listed as a Varley. If I'm wrong on that, then again, as the other trees state, the birth simply wasn't registered at all. Looking at it logically again, if there were 2 Charles Hendy's born 22nd August Chatham, one in 1897 and one in 1898, what are the odds on both birth registrations to be missing?
Annette