For years I've loaded and updated apps on my iPod, currently running iOS 10, mostly using the iOS App Store app, but occasionally using PC iTunes. PC iTunes was often useful when I wanted to restore a version of an app older than the current version in the iOS App Store, which was possible when I had a copy of the .ipa file for the older version of that app.
Recently I had a major crash of my iPod, which apparently seriously jammed up something for the Amazon Kindle app, which I had been attempting to update from 5.4 to 5.5. After that crash, I was not able to finish updating the app on the iPod using the App Store app.
Attempting to do so had rendered my existing version of the app unusable. My initial solution was to use an old .ipa file (a pre-5.4 one was the last I had saved) using the free CopyTrans Windows software, which restored decent functionality (i.e., I could read my books!), even if the app's features were not as good as those in a newer version.
I had expected/hoped that when the next version of the Kindle app came out, I would be notified within the iOS App Store app, and upgrading would be possible. Apparently things were gummed up worse than I thought, since "Updates" notifications of newer versions of the Kindle app didn't even show up within the App Store app. The display was still just stuck on the jammed up 5.5 version.
When I learned that Kindle app version 5.7.1 was available, I tried to update to it, but that, not surprisingly, jammed up my existing older (pre-5.4) version. I deleted that, and tried to load 5.7.1 using the App Store app, but again had no luck.
Finally I remembered(!) that I could download apps within PC iTunes*, something I have rarely done. Unfortunately, the Kindle 5.7.1 app (the ipa file) could not be transferred to my iPod via PC iTunes due to an authorization problem.
CopyTrans to the rescue! From their web site and probably elsewhere on the Internet, you can find out more, but I wanted to give a shout out to this great tool for management of apps on iOS (actually, they have a set of tools, some free and some paid). CopyTrans may be able to help others out with management of iOS apps and data, particularly if you are having problems using iTunes PC/Mac software and/or the iOS App Store app.
Here's hoping that future versions of the Kindle app can once again be straightforwardly updated from my iPod! (3/2017: Nope, not on their own, but upgrading from iOS 10.0.2 to 10.2.1 fixed that problem, allowing me to upgrade to Kindle 5.8 directly from my iPod.)
*Note: According to https://www.copytrans.net/blog/what-you-need-to-know-about-ios-9-and-the-backup-of-iphone-apps/, starting from iOS 9, it is no longer possible to back up apps from an iOS device to a PC (or presumably to a Mac) using iTunes, nor by using the CopyTrans Apps tool. A laborious solution (if, e.g., you want to keep the .ipa file for an app version) would seem to be to download the app into iTunes, and save a copy of that .ipa file elsewhere.