Is a child emancipated if she is 20 yr lives with her mother,works and has a child? Does her father still have to pay for childsupport in nyc?

A child is not considered a child at 20.
Most states consider a child 'emancipated' at the age of 18
New York I believe goes by age 17.

Drinking is about the only age that goes by 21 most states everything else is 18.

Specific questions on what you are wanting to know may help get better answers. Like are you the mother, the 20 yr old? Do you want to leave home or your child to move out? The father of the baby?

The father has to pay child support NO matter what! You have to go to child support/family court and file.
Is he listed as the father on the birth certificate? If not you will have to get a paternity test. The court should do that.