Thanks for the answers
It's a free Wordpress theme, but the latest update completely changed the design (may as well have been a new theme). So I re-installed the old version (still learning how to make child themes!).
Now the theme is playing up and some posts often overrun the one below them. I'm going to get penalized by Google if I don't do something. The thing is, it's a site that seems to 'punch above it's weight', and I think the theme must have something to do with it - it's quite an unusual theme.
So I've found a new free theme that I think can keep the bits that are helping me in the SERPS, while working and looking better overall. I'll do as you suggest and put it on a subdomain first.