If Pokémon you've deposited in Pokémon HOME aren't showing up (registering) in the Pokédex of your mobile device version of Pokémon HOME, there is a chance that the data is not being accurately reflected in the application.
▼ How to clear the cache in Pokémon HOME
1. Open the application and tap the menu button (an icon with three lines) in the upper right corner of the title screen
2. Tap "Clear Cache" (the fifth option)
3. Tap "OK" when the message ending in "Would you like to try clearing the cache?" is displayed.
4. A message saying "Now clearing cache..." then, "Cache successfully cleared!" will be displayed. After that, please check your Pokédex.