His mid-game departure tonight was announced to the media as a "personal matter". In Farrell's postgame, he said it was a "family medical emergency" -- but then told reporters "the hope is that he plays tomorrow." Obviously we don't have the full context, but it seems like an oddly out-of-place addendum if what Cespedes is dealing with is a truly serious matter.
Regarding his family, his mother is here in the US and according to
this article (link previously posted upthread) she's in her late 40s and suffers from high blood pressure. About a dozen other family members made the arduous journey to America from Cuba and are also here. Yoenis also has a 4-year-old son back in Cuba (living with the child's mother).