In simple noob terms; No. Consider this, generally you need around 10x the computer power to even attempt soundfully emulating a previous console. The PSX has around a 33 Mhz processor, whereas, the PSP has a 333 Mhz processor, see the coincidence? (33)(10)=330/333.
According to the SDK for the SuperCardDSTwo, an EXPENSIVE flash-cart with a dedicated 360 Mhz Processor, PS1 emulation may be possible. However, that being said, I highly doubt that PSP emulation would be possible as you would need 333(10)=3,330 Mhz processing power. This VERY THOUGHT, though, is in EXTREME INFANCY.
To answer your R4 Question in entirety. R4's are the shittiest, cheapest NDS flash cards, though they may work for simple, basic needs, whatever gave you the idea that they might run PSP games? The gaming architectures are completely different, even if an emulator for PS1 games kicks off it will be a long time before it works satisfactorily.