I get the same problem and I generally do what Hazzat said to fix it.  I also find that the 'black bars' occur when a shot has a lot of the same or similar color, in your case it's white.  So just that tip, if you want to minimize the 'black bars' appearing, try to have a wide variety of colors in your sets.  It not only helps reduce the light-flicker problem but it gives the scene an eye-catching appeal to the viewer.

Legomoviefantastic,

I had the same problem, so I played around with the Logitch webcam software.

1. Open up your Logitech webcam software.
2. There should be seven icons. Click the 5th one with the two bolts (that is the setting icon)
3. There should be six icons to the left. click the icon with the owl wearing the graduation hat.
4. It should say anti flicker, and beneath that there should be: Off, 50Hz (Europe), 60Hz (North America). I click the 50Hz (Europe) one.
5. Start Brickfilming!

If that doesn't work, and you've experimented with the other options, then send me a private message.

-Supremacy

Hey guys, sorry for taking a while to respond.
Anyway, the black bars have disappeared! After playing around with the exposure and gain settings, and using a set with different colours, the flicker was gone.
Although I am curious about using full-spectrum bulbs. They look fairly expensive online, but can they be bought at hardware stores cheaper?
Thanks for all your help!