After several sessions with Apple Support we isolated the problem. I use the Microsoft Remote Desktop to communicate with a Windows run laptop in my home office from my Macs. The program is not downloaded from Apple apps (The only one for Mac to Win comms *****.) but from Microsoft's Support site. The fapp could have become corrupted but, more likely, had been superseded by an update but I do not believe I would have received a notice from Microsoft.
In either case, I deleted the app and two helper files in the Library and the movies finally were playable without an error on the Mac. Please note that we had previously trashed the app with no joy and succeeded only after we trashed it again but with the two files in the Library. I downloaded the app from Microsoft, performed a clean install and everything is working perfectly.
My suggestion is that if you have any app that shares another screen or monitor somewhere then you may wish to uninstall it and reinstall a clean copy.
This is, obviously, only a limited solution for this error.