This appears to be due to the new security mechanism in Mojave & I have been able to 'fix' it on my machine.
There are many posts on the web that describe booting into safe mode and using a command to turn off the new mechanism. I did not do that, but rather used a brute force method. In hindsight, I probably should have tried it...
In Time Machine, under Options, you can exclude things. There are four folders in the main HD on my mac. I excluded the contents of each, not by adding the top level folder, but by including those directly under them.
For example, instead of putting /Library in there, I put /Library/Application Support and all of its siblings. Then I did the same for the other three folders.
Then, I removed the items in the exclude list one at a time. It took a while, but I found those places that were causing the problems and either updated their permissions or kept them in the exclude list until the end and then changed the assigned permissions.
Note that there were a few times that I needed to go one level deeper. That only happened on a couple of folders, though.
Also, some of the permissions changes needed were made obvious by the exclude list in that some folders had list red circles with negative signs. I made it to that I had read access to those & that problem was solved very easily.
It turns out that the fault isn't Apple's, or at least not completely. The files that caused the inability to backup via Time Machine was due to another vendor's (Waldorf Largo and PPG Wave 3) software that I had not updated recently. It seems the clash between Mojave expectations and that software's *something* caused the failure.
So, updating that software in addition to what is described above resulted in my being able to perform full back ups with no problem.
Total time spent was about 3 hours, but it works now.