It's not commonplace that someone wants to remove a father's name from a child's birth certificate. There are only specific reasons as to why a father's name can be removed, and it typically can not be removed simply because your family is going through tough times or divorce. However, it is not impossible (although not probable) to have a father's name removed from the child's birth certificate.
Make an appeal in Family Court to have the father's name removed from the birth certificate. You will need to have specific reasons as to why. These reasons can be that the father is not the biological father of the child (with medical DNA proof), or possibly because the father has abandoned the child. A judge will almost always allow you to remove the father's name for biological reasons, but not necessarily due to abandonment.
Receive the court order stating the father's name must be changed on the child's birth certificate.
Take the court order to the state's department of records (or where the court specifically tells you). Present the court order to the state clerk's office.
Wait for the clerk's office to adjust the records. They will change only what the court order states should be changed. You will receive a new copy of your child's birth certificate.