How do I repair damage to this ‘paneling’?
If that is a painted surface, one approach would be


Cleanly cut out the damaged area using a utility knife, jab-saw or similar tools.
Clean up the edges. Perhaps using sandpaper.
Fill the void using expanding foam.
Cut-off the excess foam using something like a filling knife.
Apply a surface filler and allow to harden.
Sand absolutely flat. 
Paint.


If that isn't a painted surface you'll probably never get an invisible repair without replacing the whole panel.