Continuity mistake: We first see from the inside of the door (and therefore backwards) the instructions that Mr. Miyagi leaves for Daniel to paint the house. In the next shot we see the same note from the outside - from Daniel's viewpoint. Clearly they are two different notes. The left & right arrows switch directions, the drawings of the hands change, the position of Miyagi's signature changes, and the placement of some of the exclamation points changes.
Continuity mistake: In the beach party scene when the boom box is thrown at Daniel he catches it with the speakers facing away from him, but when it was thrown the speakers were facing him.
Continuity mistake: At the beginning when Daniel is playing soccer on the beach, he has a huge sweat stain on the back of his shirt right before he runs past Ali. Moments later, when he runs away from Ali, the sweat stain has drastically reduced in size.
Continuity mistake: When Daniel first moves into the apartment, he kicks the outside door open. Then we immediately see Daniel from the inside, and the positions of the pedals on his bike has changed.
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.