Martin, sorry for the delay responding.
Tweeting sometimes seems to break the padlock (although as the problem is intermittent, I’m not sure about that). The only thing that seems to work reliably (for a while) is removing the widget then re-adding it.
Yes, I do have WP Super Cache installed but I have tried deleting the cache to no effect.
I have done a search and replace on the whole site to remove any reference to https://
Any other ideas?
I’m happy to give you access to the site if that would help?