The PSP is more than 10 years old now, and time is only marching on. As a result, the Lithium-Ion cells in most PSP batterys are all but dead now. This means it is now a rare search to find a PSP battery that is still in a functioning condition, that can hold enough charge to make the PSP actually portable.
The most obvious option is to buy another PSP battery, but this ideal is a lot more complicated than it lets on. Buying a used PSP battery is never a good choice, as all the OEM batteries are as bad as expired milk now. New official batteries are no longer being made, so the only other option in this range is to buy a third party battery. This is a rocky road, as a majority of batteries available to buy from third party vendors are either a much lower mAh than advertised, or have died in the same way as official batteries.
A cursory search around your favourite eCommerce site will unveil a wealth of portable chargers designed to keep your phone from dying from that poxy battery life they suffer with. Thankfully these battery Titans can charge the faithful PSP! If your battery holds small amounts of charge, you can keep these handy to top up the juice. If your battery is as good as dead, you can rely on the pack as the sole power source, as the PSP will treat it as an AC power source.
If you are still a keen user of the UMD drive, you may wish to combine this method with a battery. Redditor kc0nlh pointed out that when the UMD drive is active, the power draw is higher than most conventional phone charging packs. This means that when the UMD drive is in use, the system cannot allocate enough power to keep it going, resulting in the whole PSP powering down. If you run your games off the flash card, you should not experience any issues with this method.
Sadly, these chargers come in bulky cases that can reduce the portability of the PSP console. But with some handiwork, you can easily attach these packs to the back of the console. That way you can still use it as a handheld, albeit with more power and more bulk.
Restore Battery Cells
Lots of people on the PSP Reddit have attempted and successfully implemented a battery for the PSP to use in lieu of the official battery. They discovered the battery has a relatively simple construction, allowing batteries to be daisy chained to the official one so long as they provide a close match voltage that the PSP can register. The caveat with this, is that the chained battery will fall outside and require modifications. However, the best I have seen so far is that if you do not use the UMD drive, you can remove the UMD player and store a larger capacity battery in place.
There are many guides available, however we have yet to attempt any of them. Remember that if you decide to tackle this, that you are dealing with a battery. Please take extra precautions not to shock yourself on the battery, or short it causing damage to your PSP.