Let me copy from one dictionary:

Chinaman (n., pl.)

1. Usually Offensive. a Chinese or a person of Chinese descent.

2. (l.c.) a person who imports or sells china.

3. (often l.c.) Political Slang. a person regarded as one's
benefactor, sponsor, or protector: Example: to see one's
chinaman about a favor.

4. "a Chinaman's chance,"Usually Offensive. the slightest
chance: Example: He hasn't a Chinaman's chance of
getting that job.
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I thought that guy was using the meaning of 1,
but 4 is also very interesting.