Hi again, @mickee
You wrote:
If something in your Ubuntu MATE 22.04 system / installation / setup is still requiring "polkit-gnome-authentication-agent-1" besides "polkit-mate-authentication-agent-1", then I guess (may I may be wrong!) that you may try to install the "policykit-1-gnome" package (by running the command sudo apt install policykit-1-gnome) which is the package that provides that "/usr/lib/policykit-1-gnome/polkit-gnome-authentication-agent-1" file.
I've found the following related and useful (but relatively old - from 2019) discussion here in the "Ubuntu MATE Community" (started by @mate2go and replied to by @Norbert_X):
For what it's worth, I don't have the "polkit-gnome-authentication-agent-1" stuff in any of my two Ubuntu MATE 22.04 installations (one is a fresh install; the other installation has started being Ubuntu MATE 18.04, then later upgraded to 20.04 and later upgraded again to 22.04). I only have the "polkit-mate -authentication-agent-1" stuff:
$ ls -l /etc/xdg/autostart/pol*
-rw-r--r-- 1 root root 7796 dez 11 2021 /etc/xdg/autostart/polkit-mate-authentication-agent-1.desktop
$ apt list --installed | grep -i 'pol[ki]'
WARNING: apt does not have a stable CLI interface. Use with caution in scripts.
gir1.2-polkit-1.0/jammy,now 0.105-33 amd64 [installed,automatic]
libpolkit-agent-1-0/jammy,now 0.105-33 amd64 [installed,automatic]
libpolkit-gobject-1-0/jammy,now 0.105-33 amd64 [installed,automatic]
mate-polkit-common/jammy,now 1.26.0-1 amd64 [installed,automatic]
mate-polkit/jammy,now 1.26.0-1 amd64 [installed]
policykit-1/jammy,now 0.105-33 amd64 [installed,automatic]
policykit-desktop-privileges/jammy,jammy,now 0.21 all [installed,automatic]
polkitd/jammy,now 0.105-33 amd64 [installed,automatic]