I have looked on the internet and have come to the conclusion that Ubuntu-Mate 22.04 does not have a GUI element for easily setting the DNS server(s). However, some other desktop-environments do.
Furthermore, my investigations suggest that on Ubuntu-Mate 22.04 systemd looks after DNS server settings, and that changing any systemd DNS server settings via the terminal is a daunting task with a serious risk of things going wrong.
Have I overlooked something?
Is there an easy way to change DNS server settings on Ubuntu-Mate 22.04?
The "Additional" made me ask myself, "If this box is for additional DNS servers, where is the primary DNS server defined?"
If I understand what you are saying correctly, then:
The primary DNS server will be assumed to be at the Gateway IP-address.
When manual entries of DNS server addresses are made in the field "Additional DNS servers", they will override the primary DNS server assumed to be at the Gateway.
If I enter the IP-address of my pi-hole as the first item in the field "Additional DNS servers", followed by a comma and then the IP address of my Gateway, then:
When the pi-hole is active (i.e. running) the pi-hole will be used.
When the pi-hole is not available (e.g. switched off), the DNS server programmed on my Gateway will be active.