Fixed it.
My problem was that the update for the core plug-in became available before the “pro” version. So only one of my two “Events Calendar” plug-ins updated to Version 4.0, causing the message. At the same time, my license code would not “stick” so I couldn’t update to the pro version anyway. I just tried entering my license again, then saved it and this time it worked. So when I clicked on “updates” I finally got the Pro version, which I’ve now installed. And the message finally went away!