Hi there, Amnem.
I see that your Logitech mouse isn't waking so well after your Mac wakes from sleep or the screensaver engages since upgrading to macOS Catalina, and I'm glad to help.
Firmware Update Tool - Logitech Support & Download -- Make sure the firmware for your mouse is updated. Click "Show All Downloads" and select "Choose OS" under Mac. Select macOS 10.15, then "Download now".
Using a Bluetooth mouse, keyboard, or trackpad with your Mac -- If updating the firmware in your wireless mouse hasn't resolved the issue, review the information and apply the steps provided in this article. Be sure to try to unpair then re-pair your Bluetooth mouse with your Mac.
Take care.