Mixed-nationality kids born in China, where either the mom or the dad has a foreign passport and the other parent has a Chinese passport, automatically acquire Chinese citizenship at birth.
China doesn't recognize dual citizenship, so the status of these kids is classified as a "nationality conflict" (国籍冲突), and they cannot obtain a Chinese passport to travel. Further complicating things, however, they also cannot leave the country using their foreign passport, since that one does not have a valid China Visa.
So, there's a bit of a runaround to deal with this issue, but the solution in brief is as follows:
You need to get your child an Entry-Exit Permit (出入境通行证) to leave the country for the first time. The Entry-Exit Permit is usually valid for one exit and return entry within 3 months of issuance. Then they come back in on a China Visa or a Travel Permit.
Here's a step-by-step guide on how to obtain the Entry-Exit Permit. The official website for the process (in Chinese) is right here. It usually takes 7 workdays from application to receiving the document