The game is asking me to update, but I don't see one available

If the store is not correctly showing the available updates, there are some things you can try.

On older iOS versions you can open the App Store and click on the updates tab until it forces an update. On new iOS versions you can open the App Store, click on your profile button at the top right and then pull the screen down until you get the update spinner at the top.

On Android devices you can open the Settings, go to Apps and locate Google Play Store and Google Play services from the list, and from their Storage menu select Clear cache.

On Kindle devices you can open the Appstore and choose App Updates from the menu on the top left corner. If the update is not shown on this list, search for the app manually and click on the Update icon from the app page.

If the store is giving you some kind of error message when trying to update, please check out the support page for the store to find out further instructions on fixing the error:
iOS - https://support.apple.com
Android - https://support.google.com/googleplay/answer/7513003?hl=en

Kindle - https://www.amazon.com/gp/help/customer/display.html?nodeId=GRBHRJHSSV9RA2PM