Warning! Do not run your HX-20 from the power supply. The power supply should only be used to charge the battery pack.
It’s possible that a new owner of an HX-20 will not have the original Epson HOOAAG 6V 600mA DC Power supply included with their unit. This is how it was for me, so I had to find a replacement.
Surprisingly (to me), replacing one of these adapters is not so simple. You can’t just purchase a 6V 600mA power supply off the shelf (see here and here). Instead, you need to purchase one that is at a higher voltage (9V) but still 600mA, otherwise it will not charge at an appropriate speed, or potentially at all.
The power supply also requires the polarity of the connector pin and barrel to be minus (-) for the pin, and plus (+) for the barrel. If these are reversed the battery pack will not charge. Lastly, the size of the connector is important too, the dimensions of the original are 2.0 x 5.5mm, so try to match this or get one that is very similar in size.
Trying to find a power supply which fits the above requirements was somewhat tricky, but, luckily I came across this unit. It comes with several different connector sizes (included one that fits properly), the polarity for the pin and barrel can be reversed at will, it outputs 600mA, and the voltage is adjustable.