At someone's suggestion on this forum I decided to try LinuxMint on my machine that I use for trying other distros. LM has updates almost every day just like Z16. But they WORK! I have not had any problems with the update function on LM. I wish the update function on Z16 could be made to work as well as the one on LM.
There have been other threads about the updater included in Gnome-Software and its quirks. And it is noteworthy that the gnome-software updater is not the same as the System Updater included in Zorin OS.
The best way to save yourself a headache on this is to run the command @Adithyansm posted above in terminal. Whenever you feel like it, really...
I do it often because my participation on the forum makes me have to do it all the time, testing things and such for other uses. In this advantage, I am spared of having to make a routine of it.
But you can make a routine of it, or just open Software & Updates and set that to automatic updates, too.
I did run the terminal commands and that did solve the problem. I have auto update turned on but it says that is for security updates not in general updates. I get from time to time a notification that I need to install updates. When I click on that notification I get the screen I posted a screen shot of in my first post. I then click on update all and get the message you see in the small black window. Also if I just click on the software store to see what I might want to install it shows I have updates there also. But when I try to update them I get the same message. Running the terminal commands has fixed it for now but I am pretty sure I have done that in the past and after a while the problem comes back again. I will bookmark this post so I can come back to it for the terminal commands if the problem returns.
I hear you Carroll. I get them too. Sadly, it's not a Zorin issue, but they will
be blamed...lol I try not to use snap or flatpack whenever possible. But sometimes, ya just got to.