I'm going to assume all this happens during the Southern Song, after the Mongol conquest of the Jin Empire but before the conquest of the Southern Song itself.
If the Song Chinese colonized California, then its inevitable they would push into Mexico, and perhaps to South America. The incentive is in one word, silver.
Silver was the standard currency of the Chinese monetary system. It was highly prized and had a higher purchasing power in China than almost anywhere else. Spanish Americas happen to be the world's most productive silver producers.
Assuming the Chinese alchemists develop the mercury amalgamation process, I think there would be a tidal wave of Chinese silver prospectors to the Americas. Silver mining, unlike gold mining, tend to produce lasting settlements because its more labor intensive, require extensive ore processing infrastructure, and take a lot longer for the ore to run out. So these silver towns would likely grow into cities instead of vanishing like gold towns.
A sustained infusion of silver to China would be deepen the money supply. Since taxes was paid in silver, a drop in silver prices would be the equivalent of lowering taxes. This would greatly stimulate the economy which was exactly what happened during the Qing dynasty when Spanish silver flooded China.
Furthermore, the Chinese colonies would bring back a wealth of New World crops, like potato, corn, various types of squash, peanuts, chili pepper, etc. This would mean a faster population growth which leads to more immigration. Again, similar to what happened with the European colonial experience.
Should the Chinese settle Mesoamerica, then the Caribbean and Gulf cost of North America becomes accessible and attractive future agricultural colonies. So by the time of the European arrival, territorial wise Chinese Americas would look similar to Spanish Americas. Except the Chinese would have colonized Oregon territory earlier on and ignored the Midwest territories east of the Rockies. Politically the Chinese would likely rule with a long leash, satisfied with tributes of silver and jade, and the adopting of Chinese cultural practices. The Mesoamerican and Peruvian peoples would likely become Sinified and adopt to Chinese style agriculture.
The European colonies would likely focus on temperate eastern seaboard of North America. Then it would expand south and the first confrontations would likely be in the Caribbean to drive the Chinese out. Unlike the Spanish in OTL, the Chinese colonists would not be sugar plantation owners but rather poor rice farmers. It's likely the still nascent Caribbean colonies would not be fiercely defended.
Because of the relative slow speed of the Chinese expansion into South America. The indigenous populations would a have better chance of developing resistance to Old World diseases and maintain a very large indigenous population base. Over time the South American colonies would break away to form Sinified nations independent from North America while the earlier colonies in North and Central Americas may remain a unified empire with everything east of the Rockies the domain of European powers.
Meanwhile, back in China itself, the enriched Southern Song may now be powerful enough to avoid being conquered by the Mongols. They may even expand southward to Southeast Asia and the Indian ocean, along the lines of and probably exceeding the Ming experience.