Hi and welcome to the forum
Being the problem in the last part of disc (i.e. in the most external ring), my guess is that you should burn these discs at a slightly slower speed. Usually, 16x certified discs give better results if burned at 12x or 8x (neither slower nor faster).
A firmware update indeed should help to improve burning quality. Did you try to search at Sony web site to see if an update is available?
btw, Sony branded media are not exactly the very best, so I also suggest to buy better discs like Verbatim