First you will need a pin out of the adapter.
Tools you will need are.
1: Solder with rosin core thin solder would be best
2: Soldering iron lower watts the better
3: Flux to clean the surface you want to solder to
4: Sponge to clean the tip of solder iron
If you don't have any of thous supply's it would be cheaper to buy a new cord. Also if you don't have vary much experience soldering i would also recommend getting a new cord for the fact you can cause more damage than good.
To get the solder iron ready you need to heat it up. Once it is hot enough to melt the solder apply the solder to the tip and let is set there until you want to solder. You have to clean the tip off with a wet sponge but not soaked in water just enough to get it wet. Every time you go to solder Clean the tip off and apply fresh solder to it not a lot just enough to cover the tip when you want to solder. When you go to store the iron apply a small blob of solder to the tip to keep it from oxidizing.
Strip the wire shielding off a bit longer than the size of the contact you want to solder. Twist the wire and put flux on the surface you want to solder and tin them.
Tinning the wire is just applying a small amount of solder to the wire as to hold the strands together and make it easier to transfer the heat and solder to the contacts. Now mach the pin-out to the weirs to the contacts and put flux on the contact you want to solder to. Clean the surface by applying flux to it.The surface will be oxidized and will not let you transfer heat or solder to the surface. The key to soldering is to use the right amount of heat , solder and time you hold the iron to what your soldering to. You want to make sure you touch the wire and the contact you are soldering to as to transfer the heat to both items at the same time to make the solder flow. Put the solder wire to the tip of iron that is touching the wire and contact and not on top of the iron tip. Any time you apply heat to solder joints you want to use flux to keep them clean and free of oxidized material. After you solder check the solder joints for cracks, uneven and dull surface. You want the solder to look as shiny as possible and smooth. If it looks like it has waves or holes and cracks it means you have a cold solder joint and must reheat and apply flux with a small amount of solder to fix it .Here is a link to help you out this is not my site and full credit goes to the owner of site. How To Solder Soldering Tutorial
Also put a link to a new cord if you decide you want to get a new one. Here is a link that may or may not help with the pin-out. Also not mine it should credit the poster on the forum as he did all the work i just found it PSP Go Cradle mini USB Tutorial with Partial Pinout PSP Go Acidmods com