Not used the HV20 per se, but used many miniDV palmcorder types back in school days. What you are describing does sound like tape whir/whine. This is the problem with smaller consumer-type cameras, even really good ones like the HV20. So compact the mic is right there next to the tape drive...
First, if you have the ability in your menu system you can turn off auto gain. This will only make your audio pulse and pump as levels change. Sounds awful. Second, if you can afford it get something like a BeachTek box. This device screws into the bottom of the camera on the tripod mount, it has an 1/8 inch jack to plug into the mic in on consumer cameras, and has 2 XLR inputs with the appropriate controls. About $100 US and work very well. Used with an affordable shotgun mic like an Azden it will significantly improve your audio. Poor audio is the one really easy thing that separates good movies from crap ones (speaking as someone who has made a fair few crappy audio crap movies). Of course you could also go to the trouble of getting (wireless) lav system. But not having used one I cant say. Not sure if they have XLR or 1/8 connections. So yes, an eternal mic highly recommended. adam