Revealing mistake: When Pikachu meets Puka on the cliff, we are treated to an aerial shot which shows the ocean below. There is sunlight reflecting off the surface, but as the camera moves, the lights pass in front of the two Pikachu on the cliff. (00:07:26)
Hanecco's Jealousy! Battle in Big Prairie!! - S3-E21
Continuity mistake: When Ash says, "There's a couple of Gloom over there!" Pikachu's mouth is open, but in the next shot, it is closed. Also, Misty is not smiling, but in the next shot, she is. (00:01:29)
Happinas's Happy Nurse! - S3-E13
Character mistake: When Jessie asks if she can go to Pokémon Nurse School to become a nurse, the flyer she holds misspells Pokémon Nurse School as "Pokemon Nurse School." (00:07:53)
Revealing mistake: When Ash brings his first piece of meat into his mouth, another one disappears into thin air. (00:09:09)
Pokemon, I Choose You! - S1-E1
Continuity mistake: When Gary is first talking to his crowd, there is a woman with blue-green hair, a pink dress, and a yellow apron to the left, but in the next shot, she is replaced by a boy in a green shirt with brown hair and without an apron. (00:05:11)
Abra and the Psychic Showdown - S1-E22
Continuity mistake: When Ash notices the Saffron City Gym, the buildings in the background change. (00:06:38)
Character mistake: When first showing the tournament from the front, a sign misspells Pokémon Number One as "Pokemon Number One." (00:07:46)
Fushigidane and Fushigidane! Regaining the Monster Balls!! - S7-E34
Other mistake: The English title card misspells Poké Balls as "Pokéballs." (00:02:41)
The Fourth Round Rumble - S1-E77
Character mistake: When a poster for the Pokémon League is seen, it misspells it as "Pokemon League." (00:20:22)
Mystery at the Lighthouse - S1-E13
Other mistake: When Brock turns his head to the costume, his nose is missing. (00:12:47)
Suggested correction: Agree with the other correction. Dropping details, such as facial features like a nose or mouth, and simplifying character designs is a very common and encouraged drawing/animation technique to imply distance. This isn't something the animators would just overlook. It was a deliberate decision. Ash and Misty similarly look somewhat less detailed than they do in closer shots. At best, this would be a deliberate mistake, but I personally think it falls under the umbrella of "suspension of disbelief" and shouldn't be considered a mistake given it's a stylistic choice to imply distance.
Suggested correction: His nose isn't seen because he is far away from the camera.
Island of the Giant Pokemon - S1-E17
Other mistake: In the recap, when Ash, Misty, and Brock are worried about the cyclone approaching, the edges of Meowth's ears are mistakenly colored tan instead of black. (00:01:16)
Other mistake: When Ash, Misty, and Brock are worried about the cyclone approaching, the edges of Meowth's ears are mistakenly colored tan instead of black. (00:20:08)
Tentacool and Tentacruel - S1-E19
Revealing mistake: When Brock says, "Ah! Zubat!" his neck is not fully colored in. (00:15:50)
Continuity mistake: When Brock says, "Looks like a greenhouse," he is holding the straps of his rucksack, but in the next shot, he is not. (00:04:08)
Who Gets to Keep Togepi? - S1-E48
Character mistake: At the beginning of the episode, the sign on the Pokémon Center misspells Pokémon Center as "Pokemon Center." (00:01:13)
Continuity mistake: In one shot during Randall's flashback, the stripes on his shoes disappear. (00:09:22)
Continuity mistake: When Ash and Randall are about to attempt to break the pod Pikachu is in, Randall does not have his hat, but in the next shot, it appears. (00:17:27)
Continuity mistake: Before Ash, his friends, and the kindergartners throw their balls into the air, one of Ash's gloves disappears. (00:11:33)
Audio problem: After James says that Team Rocket should capture Sylveon, the sound of Wobbuffet's Poké Ball opening can be heard, despite him already being out. (00:06:27)
Giving Chase at the Rhyhorn Race! - S17-E7
Continuity mistake: When Ash and his friends look at the Rhyhorn, Ash's gloves turn completely black. This also happens when he puts on his uniform. (00:07:09 - 00:09:46)