Hi,
which battery are you talking about?
I could only find 3.7V PSP packs in the web.
Do you want to connect it directly, which means inside the c3k or are you gonna use the DC connector at the back?
If you use the DC-input, let me notice something about input voltages in general.
Normally, handheld devices can handle voltage levels +10% above and -10% under the nominal voltage level at their DC input connectors and i think it's the same with the c3k devices from sharp.
This means a voltage up to 5.5V should not damage the device.
But be aware, that this the absolute maximum value!!!
Take care of this fact if the battery is fully charged and may deliver a higher voltage level.
If you are unsure, you could drop the voltage by a schottky diode in series to the battery .
Greets,
scholbert