This is behavior related to the Contract/Confucianism auto-found bug, offered in case it can help solve that issue: When Confucianism founds in China after Contract has been researched elsewhere, a Chinese city is IDed as the holy city. However, you are not able to build the holy site (or whatever it's called) with the Great Prophet. The same thing happens even if you discover Contract before it auto-founds. Apparently, it's enough for one other civilization to beat you to Contract for them to steal the holy city, even if China is supposedly given it.
Another possible issue: Museums. Is there a condition beyond discovering Heritage [or whatever tech it is] for building Museums? I discovered the tech, but none of my cities could build them. They weren't greyed out -- they just didn't show up as an option.