Just step into any English School in Zhenzhou with your Chinese boss, and ask for help Chinese English School boss. 

You will not get it,

sorry but the business has to be registered and have a permit to hire a foreign worker.

If the business has no idea, then most likely they are not allowed to hire non chinese workers.

Next the business will need to supply you the following.

1. copy of their permit to hire foregn worker

2. a letter of invitation

3. They will need to take you to down town Zhengzhou to the one hospital that can do the health check and get a health check booklet. ( takes about 3 hours for medical) and they have it done in a week.

** then after they got all the paper work is the hard part, you can not get a Z visa in China, you have to leave the country,  the one place most people go is Hong Kong, they are issue them still, but also decline a lot also  If you go to Hong Kong, expect to be there a week, whle it process.  Regardless of what others say, use a Visa company in Hong Kong, the week we were there ( had to go twice, i got turned down first time)

By strict rules you are suppose to return to your home nation to apply for new visa, but as of last month, they were still doing them in Hong Kong