Well, it's kind of an obvious suggestion, but I'm sure a couple of days and nights in Shanghai could keep you interested, if you haven't been there before. If you are interested in more recent history of China (last 100 years), I think a couple of days in Nanjing would be rewarding, and it's a pretty decent city, as big Chinese cities go. Suzhou is between Shanghai and Nanjing, and is important for traditional landcape/gardens and high-quality textile arts such as silk and double-sided embroidery. Any of these could be worked into an itinerary that already has Hangzhou and Beijing on it. Just don't try to cram too much in.
With only a total of 9 nights, you'll want to keep to locations that have excellent, plentiful, and fast transport connections.

I think first settle on how much Beijing time you need--especially if you want to add some exploration of the ancient and/or contemporary art scene (of which Beijing is indeed the center of gravity for China) on top of the regular "biggie" tourist sites. I again reiterate, 3 nights in BJ likely won't be enough. To maximize efficiencies and avoid backtracking, I would fly in/out internationally at different Chinese cities if you can.