Most of the time you can take it to the bank of china and they will make sure it's not a fake, they will then issue you an unmarred one. Some shops like supermarkets will still accept them, but I've had troubles in the past with small shops

tape it back together and give it to someone else...

Tape it back together, put on your best "stupid laowai" face, and keep offering it to the same business no matter how many times they refuse it. Not only will you get to spend that broken note but you can be proud of yourself for out hard-heading the shop owner.

Sorry I have to say; it should be "What can you do with a bank note that has a small part missing"? The first time in over 4 years, today I had a 10 RMB note with the corner missing (small finger nail size) , the guys blew up a bit, but took it because they know me. I am usually very careful with money, but this slipped through and into my wallet!!

Well you can roll it up and still snort a line or two

I took one this afternoon into the Bank of Shanghai and they changed it immediately. No paperwork in China! I'm not kidding

When i was in bank of china i saw the cashier retaping a 100 quai note that was in lots of small places back together.  I have taken note back once and i had a burned note that i gave to my local friendly shop keeper who took it.

buy the invisible scotch tape and tape one side and pass it off, tape side down