Site problem … related to Wordfence?

Randomly, but about once a day, my site presents with broken images — header, social media icons, etc.

Then, trying to navigate will cause browsers (Safari, Firefox, Chrome) to throw a warning about the site not being secure. After a while it resolves … until the next time. Of course, this is not good for my reputation. It’s been going on for several weeks and my hosts claim it’s not an issue with their servers. However, I’ve noticed I get the same warnings when I try to access stats from the hosting cpanel — that the certificate is not trusted or something.

Wordfence has not detected any hack (though, in the past, I’ve had warnings that some licence.txt or readme.txt has changed, which I’ve ignored).