It was just a matter of time before Retroarch would be able to play PSP games on the PlayStation classic.
This PPSSPP core has been tweaked and optimize to run on the PlayStation classic. It said that the PSP core runs better on the PlayStation classic than the NES classic or an even a Raspberry Pi 3.
Now results may vary depending on what game you are running with the core. Speeds should generally be good with some graphical glitches expected. If you experience any graphical glitches but your game runs well you might want to try setting Block GPU on. It will reduce performance but it will also reduce the graphical glitches. There are also a number of various core config settings under the RA quick menu which you might want to try adjusting to get the settings tweaked right for each of your individual games.
Installing the core just requires dropping on top of your preexisting RetroArch setup how-to videos are available below.