Hi!
So the power switch won't move anymore? If this is the case, open up the PSP go and check if there is anything inside, also clean it there.
If nothing is wrong so far and it is cleaned, then it need to be replaced. This repair will not be easy as you need to solder the new power switch in.
Part: https://kasynparts.com/product/on-off-power-button-switch-for-psp-go-replacement/
Guide: [guide|1185]
In step 16 (2nd photo) you will find the power switch (left) soldered on the motherboard. Desolder it and solder the new one in again.
Please let us know how it goes, and good luck!