IF YOU'RE RUNNING CATALINA, then thereal issue is likely related to its new security features. You don't need to install any additional software. You just need to follow the steps below to make sure everything's set up right. Unfortunately these installers don't quite "get it right" on Catalina.
Go into System Preferences >> Security & Privacy
Click the Privacy tab.
Select Accessibility from the left pane
Check that Logitech G HUB is CHECKED
Make sure Logitech G Hub is checked for Accessibility. If it's not already checked, this is likely your problem. Check it, then proceed to step 5.
Exit, then re-launch the G Hub app.
If it still shows disconnected, unplug and replug mouse
If it still shows disconnected, uninstall and reinstall G-Hub (I had to do this)
a. Exit G Hub (obviously)
b. Open Finder, go to Applications folder
b. Delete the Logitech G Hub app (move to trash)
c. Re-install G Hub the same way you did before.
You're mouse should show up correctly now and let you customize, etc. The key here step 4, which is just one of those things that are now required with Catalina. If you're lucky you'll only have to go through step 5. In the worst case, you might have to do all steps through 7c. A reboot somewhere between steps 5 and 7c might actually make it work, too, but I'm only listing the steps I actually performed - I hate rebooting.
Hope this helps!