A Custodial Power of Attorney (CPOA) is a document that gives another person temporary power to make decisions for your children. It is advised that you create a Custodial Power of Attorney with your designated caretaker as a way to possibly avoid having your children go into foster care if you are picked up by ICE. A Custodial Power of Attorney is a private agreement between you and another person, and is NOT a court order. If the children's mother and the father both live in the home with the children, both parents should sign the CPOA.
This interview on immi.org helps you to understand your immigration options. It has questions about qualifying for a way to stay in the United States. It does not screen for all the ways to stay. It asks about your eligibility for: family-based immigration, humanitarian options (asylum, Temporary Protected Status, Special Immigrant Juvenile Status), U or T visas for victims of crime or trafficking, VAWA for abused family of citizens and green card holders, and DACA (Deferred Action for Childhood Arrivals)