Andrew,
since it's that goop on the screen two things I'd try.
Gently use a bamboo skewer to try picking off some of the bits and then wrap the tip of the skewer with a small bit of cotton from a Q-tip dipped in alcohol and squeezed almost dry. The key is all about control and patience.
I've also used dental probes with an offset tip to pick that stuff out of the grooves. Wasn't perfect but didn't have to replace the screen either.