Purged and reinstalled.
Same.
Says:
HPLIP cannot detect printers in your network.
This may be due to existing firewall settings blocking the required ports. When you are in a trusted network environment, you may open the ports for network services like mdns and slp in the firewall. For detailed steps follow the link. https://developers.hp.com/hp-linux-i...leshootNetwork

When trying via the temporary usb connection it names the printer and then says:
An I/O error occurred.
Please check the USB connection to your printer and try again.
(Device I/O error)

I seem to have split the thread I think.
Anyway system-config-printer produced exactly the same deprecation rich output as before.
Graeme

#1 somehow there seems a corruption on your installation. I would back up critical data and do a clean install and recovery. You might take time to check first, this could be at the REPO level and a known issue they are working on, ot they may have a work-around to fix the issue.

#2 It looks more and more to me like a faulty printer. No amount of playing around in software or configuration will fix bad hardware. Can you test it on a different device?

It (manually) connected fine at first and worked for a year or two. I have wondered if maybe a kernel upgrade upset it.There have been a few more since it lost connection.

The printer works fine on the windows computers in the house and another linux I have. No problem with it.

I upgraded mint mate from 20.03 to 21.1 but didn't do a clean and reinstall.

The main clue is that the printer can't be detected on the network, maybe like a firewall is blocking it. I initially assumed a network problem but all else is good.
Even when I connect it by usb it recognises and names the printer but won't work.

The hp support site says:
Make sure that port 161 (udp and tcp), port 162 (udp and tcp) and port 9100 (udp and tcp) are open through your firewall. If these are not open then HPLIP will not function.
How do I do that?

That depends upon which firewall you have. If it is the distribution default then firewall settings (under some name) should be in your menus.
I often simply disable all listeners and run without firewall as long as I am not directly connected to the internet. (But I never run without malware/AV protection.) Your menu may have an option to turn the firewall off, and that should suffice for testing.

NOTE: a direct USB connection bypasses any network issues whatever. That keeps throwing me off. If it works for all of your other machines, then it must be something about this single machine. I cannot replicate that. My printer works from phones, tablets, and all versions of Linux clients exactly the same with no issues. (Except that my printer is getting OLD and starting to wear out.)

Thank you for your time and efforts.

I went to

How to Setup and Configure UFW Firewall on Linux Mint 20

after seeing in the HP Developers Portal doc that ports 161.162 and 9100 had to be open and eventually opened them.
HAPPY DAYS!!!

The printer was found automatically and works.

I love Linux. It's a real adventure and escape from microsoft's domineering. Escaping google is proving more difficult.

Thanks again,
See you with the next problem,
Graeme