Was it Package Manager that you used to install Handbrake ?
It appears the system is still set to use the stebbins PPA. A PPA (Personal Package Archive) is a personal repository for a developer to distribute software outside of the usual (Main, Universe, Restricted and Multiverse) repository system. As PPA are not 'official' they may be less reliable than the proper sources.
When you installed Handbrake, I'm guessing you followed some instructions which involved allowing the stebbins PPA to update your system. It looks like the authentication part of the instructions didn't work properly so the system is set to look for updates from the PPA but cannot actually do that so the update process cannot complete.
To remove the stebbins PPA from the list look in the manual at
https://www.linuxliteos.com/manual/software.html#pparemThen try updating again.
Some common extra application (including Handbrake) can be installed using 'Lite Software'. If you are installing extra software, generally it is best look for it in Lite Software first. If it isn't available there, try the Package Manager.
In general you only need to look for applications from 'unoffical' sources like PPA if the 'official' routes of Lite Software and Package Manager don't provide the application (or the version you need).
