Q: Recently Added songs on my iPhone aren't showing up on my iMac's iTunes Music Library.

Hello DrKpe,

It sounds like you are expecting content made available offline on your portable device to sync to your computer. According to this article you need to make them available offline on each device individually.

Add music from the Apple Music catalog to your library

You must save songs, albums, or playlists on each device that you want to use for offline listening.

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