I have several P6R receivers. It mentions in the manual to use 9-volt alkaline batteries, but will using rechargable 9-volt batteries affect or damage the receiver in any way?
A rechargeable battery will not harm the P6R receiver or any PSM receiver. But a rechargeable 9 volt battery has less current capacity of an alkaline 9 volt. Therefore, the operating time of the P6R will also be shorter. If that is an acceptable trade-off, then the rechargeable battery is an acceptable power source.
Also, see the following link:
Batteries and wireless microphones