Hello Joseph_S.
I signed into iCloud with my Apple ID and Password on my iMac desktop computer as your suggested. The sign in went well and my iCloud account opened fine and displayed all the icons for the Account Settings. My name appeared in the upper right hand corner of the window.
Next, I clicked on the phrase "Account Settings" , which appeared directly under where it said "Good Evening Gary", and a new Settings window opened. This window displayed three categories, "Storage", "My Devices", and "Advanced". The unusual thing that I noticed right away, was the fact that under the "My Devices" category, only my iPhone SE was listed. It did not list my iMac computer as one of my devices, even thought I had logged into iCloud on that computer, per your suggestion, before starting to run these tests. It is on my iMac computer that this problem is taking place, and on which I am constantly seeing the error messages displayed, which refer to a problem connecting to or using my iCloud account, in connection with my iMac.
One message says, "This Mac can't connect to iCloud because of a problem with (displays my correct user name)". "Open iCloud preferences to fix the problem".
Another says, "Photos for the iCloud account (displays my correct user name)" cannot be accessed. Review your account information in System Preferences".
I have tried to follow these instructions without any success, and that is why I am asking for your help.
How can I add my iMac to my list of iCloud devices, so that this problem is solved? I would also like to be able to backup my iMac to iCloud automatically, in the same way that I am able to backup my iPhone currently. I am running OSX 10.9.5, if that is of any importance.
Thank you.