Miyagi: Wax on... Wax off. Wax on... Wax off.
Miyagi: We make sacred pact. I promise teach karate to you, you promise learn. I say, you do, no questions.
Lucille Larusso: Look at those palm trees. Damn! You know what that means?
Daniel: Yeah. Watch out for falling coconuts.
John Kreese: We do not train to be merciful here! Mercy is for the weak! Here, on the street, in competition, a man confronts you, he is the enemy! An enemy deserves no mercy. What is the problem, Mr. Lawrence?
Daniel: This school sucks, man. Sucks.
Daniel LaRusso: So I guess you know about the tournament tomorrow.
Ali Mills: Who doesn't?
Daniel LaRusso: Dumb move, huh?
Ali Mills: Not necessarily.
Daniel LaRusso: I don't have much of a cheering section.
Ali Mills: You got me.
Daniel LaRusso: Well, I'll probably get killed in the first match anyway.
Ali Mills: So we'll leave early.
Answer: Daniel's wounds came from fighting the Cobra Kai guys but he tells Miyagi that he was injured falling off of his bike. Miyagi sees through this and sarcastically says that he's lucky his hand wasn't hurt.
Damian Torres
I think he was also subtle letting him know he knew he didn't really fall off his bike because if he had, he would have at the very least scraped his hands...especially if he fell hard enough to cause that much damage to his face.
He was still trying to hide it. However, if you fall, you instinctively put out your hands, and they would be scraped up. Miyagi, older and experienced in life, knew better, but Daniel wasn't going to admit it outright.