It's almost impossible to pinpoint the problem without actually seeing it, but if it's a new HP my first reaction would be to reformat the hard drive and perform a clean install of XP. HP installs all kinds of junkware that just hogs resources and can cause burning problems.

Since you're getting failures on both drives, the problem is likely either media, something you're doing wrong, or something wrong with the computer (see above, there is probably nothing physically wrong with it, just might need a clean install).

What speed are you burning at? Have you tried lowering the speed to see if you get a good burn?