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As Morf suggests, a little more info might help us to nail down where your problem is.  Here's some more info for you, too.

On the DX you have books loaded.  The ones actually on the device show on the home page and/or in collections and the ones that are not on the device but safely stored at Amazon are in the Archive.

On the smaller Kindle you may or may not have downloaded the same books.  It doesn't happen automatically just by buying a new device.  You have to specify, individually, what books you want to download.  Any you do not will show as being in the archive (or cloud, depending on the model).  

The two sets, 'device' vs 'archive/cloud' are mutually exclusive for a specific kindle.  But they'll be different sets for a different device.

If you are using collections, you may have imported the collections from your first device to your second device.  If so, any book already in a 'collection' on the first device, will go directly into that collection on the second device when you download it there.  So it won't show on the home page.  This may make it appear like it didn't download when you thought you'd sent it.  Look in the collections and see if it went there. . .or set it to not show a sort by collections so you can see all the books.

That's a very general explanation. . . . . with more information about specifically which smaller kindle you have we can probably be more helpful.  The file systems are slightly different, and different, too, than the DX.