Logitech Mice and the "associated" Logic Control Center can be frustrating! Myself, I use a Logitech M525 cord less mouse on my late 2012 Mac Mini, and a Logitech M325 cord less mouse on my mid 2017 MacBook Air, and the Logitech Control Center does me no good (I am also running the latest version of High Sierra, OS 10.13.6, on both machines). In fact, it is not required for me to have it (and actually interferes with operations involving the mouse). But for "some" other models, the Logitech Control Center is required. I believe the version of that software is now V3.9.7.56, released last month. Here is the link for it, in case you need it (note that it is compatible with Mojave):
https://support.logitech.com/en_us/software/logitech-control-center-for-macintosh-os-x
I use SuperDuper! for my backups, and I am not having this issue. If you either have the Logitech software, or you install it and still have issues, you might want to consider doing a clean, fresh installation of High Sierra, and then "migrate"/copy needed "stuff" from your Time Machine backup. However, not sure if there is a "problematic" file/folder on your system (and thus on the backup) that is causing the issue.