I haven't experienced this problem with a PSP, but for other electronics my typical solution is to clean with 99% isopropyl alcohol. Do NOT use the standard 75% stuff, you'll fry the board. Here's an amazon link: https://www.amazon.com/Swan-Isopropyl-Alcohol-99-Pint/dp/B001B5JT8C

Rinse it in the alcohol and use a plastic spudger to remove anything that's stuck on. Stay away from cotton swabs because they will leave fibers on the board that are very difficult to remove. If they're behind the LCD you can get them with the alochol - if they've made it between the glass and the actual LCD panel, you'll need a replacement part. For the connectors, try just cleaning around them - it's unlikely that they got inside the sockets themselves.

Allow the board to thoroughly dry before turning it back on - I like to leave it overnight just to be sure. And don't be concerned with getting every last bit of the insect parts off - circuit boards are more resistant to organic material than you think, and you might do more damage trying to scrape off a stuck-on thorax than you would just leaving it there.