Thanks for your reply.
My Mac is a Late 2013 iMac with 2.9GHz Core i5 processor.
The authorised Apple reseller/repairer advised that Mojave should be compatible with QuarkXPress 9 and 10, both of which I have licenses for.
Editing the images does not work.
I have also tried running QuarkXPress 10 and it crashes almost immediately, but on this machine it did that when I upgraded to Sierra and so I just continued using version 9.
I can see why they may have stated that, but it is more likely incorrect. The first version of QuarkXPress that was officially compatible with Mojave is QuarkXPress 2018 (Version 14): QuarkXPress Now Supports macOS Mojave and Offers Dark Mode. Older versions may work, more likely QuarkXPress 2016/2017, and compatibility falls as you go back in versions.
I haven't used QuarkXPress since version 6/7, so I probably can't give you any viable suggestions. You could contact Quark, but they will probably state that they don't officially support version 9/10 in Mojave.