[Radioroslyn]

Great! Sound like your making good progress. You can give the 1000mf cap a try to hear if it lower the hum a little.

I suspect that the dc out of your charger causing your HV to be low. The output of the charger is pulsed dc, in other words every one 60th of a second there is no voltage at the 6v output. The filter cap charges up when there is a 6v pulse then where it drops to zero on the negative cycle the cap will discharges and fills in that (negative) no voltage cycle.

Think of it like this you are walking along using both feet and the walking is nice and smooth. Take one of your legs and tie it so your foot is up in the air. Now you can walk but it's not so smooth rather jerky. This because you aren't able to use both feet only. The filter cap acts like your other foot to smooth out pulses. It holds up the dc voltage when there isn't any.