The birth certificate correction form cannot be used to add/remove a father to/from the birth certificate.  
Add a father/parent to a birth certificate
To add a father to a birth certificate where a father is not currently listed, complete and submit the Voluntary acknowledgement of paternity form | In Spanish.
Remove a father/parent from a birth certificate
To remove a father's name from a birth certificate, you must submit a copy of certified court documents showing paternity determination with a completed data worksheet for the new birth certificate.
Court-determined paternity documents are required to add, remove or change a father's name on a birth certificate when another man is shown as the father of the child on the original certificate.
A new certificate may be prepared only when a determination of paternity is made by a court of competent jurisdiction or following adoption. A divorce decree is not acceptable evidence regarding paternity.

Report of paternity determination (for court use only).

Certified copies of acknowledgement of paternity
A certified copy of the acknowledgement of paternity can be used as evidence to establish paternity for agencies such as the Social Security Administration.