Using Kindle for Android (K4A) with CC + calibre presents challenges. The first is the Digital Rights Management (DRM) applied by Amazon to most books. The second is that the K4A application is strange, not implementing normal Android features.
1) DRM Applied to Books
Amazon "locks" most books you buy from them using DRM. Amazon's intent is to prevent people from reading books that they didn't buy. Unfortunately, the DRM can also prevent you from doing reasonable things like using CC & calibre effectively. Because Amazon's DRM is per-device not per-user, you cannot read a book on device A that is downloaded on/for device B. Said another way, you cannot download a book on your computer to be read with Kindle for PC/Mac then expect to be able to read that book in K4A without downloading it again. Also, you can read DRM-protected books only in K4A. You cannot use a different reader app. For these reasons many people remove (strip) Amazon's DRM on books they buy when the book is loaded into calibre, using a calibre plugin provided by Apprentice Alf and friends. Stripping the DRM allows you to read your book on any of your devices using any reader app you choose.
2) K4A Strangeness
K4A is one of the very few reader apps on non-Amazon devices that does not publish to Android the kinds of books that it can open, and does not provide a way for another app to tell it to open a book. These Amazon choices mean CC cannot open books in K4A. There is nothing we can do about this.
On Amazon devices, CC can ask the builtin reader app to open a book, but unfortunately doing so automatically resets the reading position to the beginning of the book. Again, there is nothing we can do about this.
You can resolve these problems one of two ways:
1) Use a different reader app that can open Amazon formats such as mobi, AZW and AZW3 and remember the reading position. There are several. If you go this route and want to sync reader positions across devices then you might want to look at our FAQ answer Problems with syncing read positions in reader apps like Moon+.
Note: You can use a different reader app only if your books have been stripped of Amazon's DRM.
2) Change CC's storage folder to K4A's books folder. This is a partial resolution in that it permits K4A to open books, but does not fix the "CC cannot tell K4A to open a book" problem. On non-Amazon devices the K4A folder's name is "kindle". On Amazon devices such as a Fire tablet the folder name is "Books" (this is CC's default on Amazon devices).
To change the folder CC uses for amazon books see How do I choose which formats calibre sends and where the books are stored? for instructions. Be sure to enable (check) the Amazon formats you want to use.
Important: if you have not stripped the DRM from your books then K4A can open a book only if it was downloaded by (or specifically for) that device. You cannot read the books on a different device, which means that there is no point in sending the books from calibre to that different device.
Important: K4A does not support subfolders in the "kindle" folder. Because of this restriction, when using K4A you must not use a save template that sends books to subfolders. Neither the default calibre save template for the wireless device nor the default template used by the Content Server and Cloud Connections use subfolders. If you have changed these templates then you must change them not to use subfolders. You might want to consider using CC's file template everywhere so you can be sure that the same file names are used by all three of the CC connection types.
When you change the storage folder, books in CC will be moved to that folder and books sent in the future will be stored in that folder. The next time K4A is started it will import the books you have sent to Calibre Companion so you can read them.