Would I need a hacked PSP to run custom translation patches?
There's currently work being done for the Puela Magi Madoka Magica PSP Game to have an English translation. It's still not done yet. However, in general, if a Japanese PSP game had an English patch for it, would I need a hacked PSP?
Solution 1:
It depends on how the translation will be released.
PSP doens't have any regional locking for games, so technically a Japanese game on UMD or downloaded from the Japanese PSN store should work on any PSP without a problem.
If the translation doesn't require modifying the game image in any way (for example, if it's an official release or theoretically a bootleg UMD or a modified save file), you should be good to go; if, however, you need a patched UMD ISO/CSO, then you'll need a modded PSP, capable of running games from outside sources.
Solution 2:
It depends on the size of the game. Games under 1142MB can be signed with a demo key to run on any PSP updated to the official 6.60 firmware and up (smaller games can be signed to run on earlier versions of the firmware). This method can be used to run homebrew software and fan translations without cracking your PSP.
The program used to sign games that way is called Fake NP, you can find it using Google.
Games over 1142MB can't be signed that way and will require a custom firmware or some other modification to run, at least until Sony releases a demo that is larger than 1142MB.