When Sayuri discusses the conversation between Dr. Crab and Hatsumomo with Pumpkin, Pumpkin says 'I sat outside the door, but I could still hear them,' then later in the description she says 'Then the Doctor looked queasy, like he didn't want to hear any more.' If Pumpkin was sitting outside the door, she couldn't have seen the Doctor's face.
When Chiyo apologizes to Mameha about the kimono; when she first bows both sleeves of her kimono are behind her arms but in the next scene when Mameha kneels beside her, the right sleeve is on top of her arm instead of behind it.
After her first encounter with the Chairman, Chiyo is shown running from the hanamachi of Gion, through the red Torii gate trails of Fushimi Inari to a shrine where she gives coins as an offering. In reality, not only is Fushimi Inari several miles from Gion, and therefore an inconceivable location for a small girl to make a round trip to on foot, its gates form a circuitous path on the side of a mountain and do not lead anywhere outside the shrine itself.
When Hatsumomo and Pumpkin are leaving on the night of Pumpkin's debut, neither Mother nor Auntie spark flint on their backs. A Geisha would never leave her okiya without this act being performed as it was believed it brought good luck.
Japanese shaved ice is a dessert that is usually served during the summer. When the Chairman buys Chiyo a cherry-flavored shaved ice, it appears to be the cherry-blossom viewing season, where it would be unlikely that any street vendors would be selling this treat.
When Chiyo/Sayuri and Pumpkin are listening on Mrs. Nitta and Mameha's discussion of Chiyo's future, and the camera zooms in on the girls' eyes peering through the crack in the door, Chiyo's eyes move very fast and you can clearly see Chiyo's blue contact, since it does not move with the rest of her eye.