PSP Go won't stay on

It is probably a case of a bad connection somewhere inside the gadget. If you are technical or you have guts to explore unknown territories then I suggest go on Youtube , find teardown procedure, check every wire and every connector in and out and if this will not be helpful enough, try to "reheat" carefully crucial elements on the board (always avoid capacitors, transistors or anything containing plastic melting material).

Last part is really tricky and needs a lot of patience. You can find a lot of articles on the internet about reheating logic boards.

IMPORTANT: STOP USING PREVIOUS ADVICE WITH BUMPING DEVICE AT THE BACK.... you are causing even further damage - in case that you have just a disconnect wire situation this loose wire can accidently touch something which shouldn't be touched and permanently damage your device ( loose metal pieces or elements are the same ). Usually if little bump is not good enough we need a bit more force, than kick, later little throw and in the end - throw in the garbage bin.. :-)