iPod touch 4 and older versions of apps on iOS 6.1.6

It's down to the app developer (not Apple) to continue to make available a version of the game that works with older iOS versions. So if the game developer doesn't do that, then you will see the message you mention.

There is no fix.

This is a simple matter of convincing the third-party app developers involved to either spend the money to remove about a decade of effort, and revert from 64-bit to 32-bit addressing and APIs, and to run ever-larger and more resource-intensive apps in an iPod touch with far too low performance (A4) and far too little memory (256 MB LPDDR), or alternatively to convince the third-party developers to add substantial parallel support costs by adding iOS 6 support for some older app in addition to current-version support efforts and costs because we all know people are going to log support requests and will expect fixes. Rebuilding and shipping an old app for iOS 6 is a project in itself, as it’ll involve installing very old versions of macOS and Xcode on an older Mac, and little or nothing (re)built by those older tools would even be acceptable as a new app store submission, which means convincing Apple to allow older versions through the app store. (Also not a big market here for the costs and the effort, as usage of old iOS versions drops off pretty quick, too.)