Way, way back around the mid-1980s, I checked out a hardback novel from a school library. I think today they would probably call it "a young adult novel." It had some illustrations, and two of the main characters were about 12 years old. (At a guess -- I'm not certain of the age, but I'm thinking "not quite teenagers yet.")
As the story begins, it is established that the third-person main viewpoint character is a girl who will be spending the summer with some relatives. Her uncle, aunt, and their son, who is about her age. (As I said -- maybe 12?) That little nuclear family lives somewhere near the beach, somewhere in 20th Century England. One reason the girl has been invited to visit them for a few months is so that she can be friendly company for her cousin, who is still recovering from some nasty illness (I can't remember what) which has left him feeling "crippled," in the sense that while he still has legs attached to his torso, and the bones are not fractured, they aren't strong enough to let him walk around without the help of crutches, and he's depressed about this because he's not sure if that will ever change. (I don't remember what the doctor's prognosis was on the chances of full recovery, or if we were even told.)
Around the end of the first chapter, the girl is alone on a beach and meets a friendly man who says some odd, poetic things. (I don't mean he was rhyming all the time.) I think he then kinda-sorta disappears, in such a way that it is at least possible that he has genuine magic powers (although he does not make this claim at that time).
A bit later, a man shows up at the front door of the family home, saying he's the special tutor whom the boy's parents have retained to help the lad spend the summer catching up on the schoolwork he missed recently when he was so sick. This man is, of course, the same man whom the girl recently conversed with on the beach. (She does not mention this to her uncle and aunt.)
From here, the novel somewhat resembles the Mary Poppins books (although written, I'd say, in a slightly more serious tone, and aimed at a somewhat older target audience). That is to say: The mysterious tutor sometimes takes the kids on magical adventures to strange places, meeting nonhuman creatures out of legend, but the adults have no idea that anything extraordinary is taking place. It becomes increasingly apparent that the tutor is more than just a mere mortal, although no detailed "origin story" is ever provided. I have a very vague idea that it may have been hinted, at some point, that he could be Puck, aka Robin Goodfellow.
Toward the end of the book, the male cousin is infuriated by something nasty that another boy has said or done, and jumps on the offender and has a vigorous fight with him . . . without the invalid boy using his crutches, or even feeling the need for them, at the time. After some adults have come along to see what the fuss is all about, the tutor says, to the boy's father, something along these general lines: "Sir, I don't claim to be one of those faith healers you hear about, but it seems to me that a boy who can hold his own in a fight with [insert other boy's name] is a lad who doesn't much need those crutches." Which seems to be true, as the invalid boy, after hearing that, manages to walk back to the house under his own power (although perhaps wobbly on his feet). It is implied that the tutor's magical influence over the summer may have had something to do with the boy's recovery.
I have a vague idea that the tutor was called "Mister Whisper" or "Mister Whistler" or "Mister Whistle" . . . or something similar, with the surname starting with a "W." Maybe. But I could be wrong! Some Googling, just now, turned up hits for various fictional characters who each have used one of those names, but none of them appear to be the right one. (For instance, a woman named Margaret Mahy wrote a children's book called Mister Whistler, with illustrations by Gavin Bishop, but judging by the reviews on Amazon.com, that definitely is not the one I was remembering.) I also have a vague idea that the word "Summer" may have been part of the title of this juvenile fantasy novel.