Something I’ve noticed for a while is that Amazon doesn’t automatically update your already-owned books if a small change is made (like typo fixes). The same is true of cover updates. You can get the covers to update inside the books themselves if you tap and hold the book cover in the library to get the menu, then “Remove Download”, then tap and hold again for the menu and hit “Download”. This shouldn’t break your notes or set you back to the beginning of the book, but it will pull down the latest version of the book, which should have the new cover at the beginning.
Unfortunately this does not update the cover shown in your library.
The only way to update the library listing covers is apparently to clear the Kindle app’s cache. This can’t be done by simply terminating the application and restarting it. And at least in current (16.x) iPadOS or iPhoneOS, there is no method provided in the Kindle app’s settings to clear the app’s cache, neither in the app itself or in the iOS settings app (Settings/General/i(Pad or Phone) Storage). (Couldn’t tell you about Android. Don’t use Android. Google is evil.)
No, to get the new covers to appear in the app’s library listings, you must remove the app from the phone or pad (which deletes all of its cached data) and then reinstall it. Which works and doesn’t seem to break anything since most of the data concerning which books you read last and where you were in those books is kept in the Whispersync cloud.
But it’s a pain in the ass to have to delete the app and reinstall it to do something that ought to be handled with a button in the app’s settings to clear the cache on demand.
I did find, at least for the Kindle Oasis (which I own), you can get the library covers to update by removing the download, re-downloading the book, and then opening the book. This apparently convinces the Kindle device to re-cache the cover. Which begs the question of why the same isn’t true of the iThing Kindle apps.