Yes that would cause the copying problems. Verbatim is very good media so I would not expect that to be the issue here. Try a burn at 8x, and update your burner's firmware to the latest available. Your HP burner is made by another company as they don't actually make burners themselves, just brand them, but I don't know what model it actually is. I believe it's a Liteon, you might want to start a thread in the Liteon forum here at CDFreaks asking how your burner can be 'crossflashed' so that you can get better performance out of it.
As for copying those discs that are poorly burned, you may not be able to, sorry to say. There are some programs available that continually re-read the disc and try to get more data off of them, but they can take hours or even days and are not necessarily going to be successful, so unless the data's vitally important then you should just update your drive, try burning a little slower such as 8x, and test your discs using the 'transfer rate test' to see if they are readable or not.