Have you tried reading with both of your ODDs?
If not please try the other ODD. Maybe even a friend's?
Often when one ODD fails another will be successful.
I doubt this will help, but you can also
untick Enable read-ahead cache in Common Settings > Read.
This essentially serves as the drive buffer and when enabled, it maximizes the rate of data transfer. Since you're experiencing read errors, disabling the read-ahead cache may limit the rate at which the read errors occur and perhaps even the number of read errors.
It's easy enough to do, so it's probably worth a shot.
You can also try to click ignore when the read error pops up, but you've probably already tried this.
If you're comfortable flashing firmware, you can look to this, but frankly, it doesn't sound at all like a firmware issue and since no other discs present problems, I wouldn't recommend it. You may be worse off afterwards.
HP burners have horrible firmware support.
Depending in the oem, you may want to x-flash, particularly if it's a LiteOn. This may be a reasonble step.