Yes, the point-based maps typically allow you to move a feature.

I can't see how to do this in the mapcomplete UI.

Take this link for example: https://mapcomplete.org/food.html?z=16.1&lat=51.400908130317... .

The restaurant "Kefi" is incorrectly located on an entirely residential road. I believe it is probably supposed to be "To Kefi", which you can see here on Google Maps: https://www.google.com/maps/place/To+Kefi+(Greek+Restaurant+...

I've tried dragging the icon, right-clicking it, or left-clicking it and looking in the popoup menu for a location field to edit, but no luck.

In the popup when clicking it there's a "Move object" near the bottom. You must be logged in for that to appear, though.