Will 6v 300mA power adaptor work with a 6v 500mA device?

Most probably you are using a AC (117V or 220V) to DC 6V converter adapter. If yes, then the answer is no, you cannot use a lower current rating for a device that draws more current. If you plug in a device that needs DC 6V 500 mA to an adapter that can only supply DC 6V 300mA, then the adapter could start sending higher than 6 volts current which may damage your equipment or overheat the adapter.

If you are using a higher mA rating adapter, then it's okay.

For more technical details, see:

http://www.jaycar.com.au/images_uploaded/plugpack.pdf