Command to find process and restart it
Here is what I'd try...  Create a bash script using the code below and save it as a plain text file without an extension.  Place it in a directory that in your $PATH, e.g. /usr/local/bin and make it executable using: chmod +x filename where filename is the name you gave the bash script.

#!/bin/bash
kill $(pgrep LCore)
sleep 1
open -a  "/Applications/Logitech/Logitech Gaming Software.app"


Now in a Terminal type the name you gave to filename and press Enter...

The $(pgrep LCore) portion of kill $(pgrep LCore) should pass the appropriate PID of /Applications/Logitech/Logitech Gaming Software.app/Contents/MacOS/LCore to kill and terminate its process, wait a second and then open the Logitech Gaming Software.app application bundle.