Is it possible to train Google Photos to better recognise specific people? I.e. if GP has incorrectly identified someone and placed the image in the wrong People album, can you move it to the right one?
Yes, although it may not be that intuitive.
Open up the "person" album where someone is mis-tagged. You can do that by clicking the person's photo in the Info pane of a photo or by searching for the name.
Hover over one photo, then choose the checkmark. Check however many other photos are also misidentifying the person. When you're done with that, open the overflow menu (three vertical dots) and choose "remove results".
That will disassociate that face from the person, giving you an opportunity to re-identify it. What you should end up with is a collection of photos for that face that don't have a name. You can then give it the right name.