Plot hole: Officer Jenny asks Ash what he knows about the "trio" when he tells her that the pirates are likely Team Rocket, but he hadn't said anything about there being three of them before that. And Jenny obviously didn't know anything about Team Rocket, either.
Revealing mistake: When Team Rocket is running away with the trophy, James falls flat on the ground, and an indentation of his body as well as the trophy is made. The indentation shows a perfect match to the trophy, but James' arms were partially covering it, so there shouldn't be a perfect indent of the trophy in the ground.
Character mistake: Misty is notoriously afraid of bugs. She makes this clear whenever bugs are around. She seems to be okay with Tracey's Venonat, but for some reason doesn't think it should be classified as a bug, when its type is Bug/Poison, and it's quite obvious that "Venonat" comes from "venom gnat, " a gnat being an insect. She even directly states that "At least they're [Venonat and Marill] not Bug types!"
Continuity mistake: When Meowth comments that Victreebel's Sleep Powder works well, James closes his eyes, tilts his head up, and says "Yes, we're a knockout combination." As he says this, his nose inexplicably grows to about ten times its normal length, but goes back to normal after he says the line.
Plot hole: When Team Rocket decides that they're going to capture Dragonite after it's been worn out, they appear in the blimp above the battlefield no less than ten seconds after they were shown in the stands. There is just way too little time for them to have possibly gotten up there that fast, even allowing cartoon physics.