If the undocumented parent has lived in the U.S. for over 10 years, has good moral character and
the children will suffer extreme and exceptional hardship if their parent is deported, then the parent can file an application for cancellation of removal. The only catch is that the parent will have to be caught by the immigration police before they can file the application.
In general, no without further information. A person who is placed in deportation/removal proceedings may try to prove that their deportation will cause their US Citizenship children "extreme and exceptionally unusual hardship." If the mother is also unlawfully present, than this causes complications in trying to get relief. However, there are more requirements that may disqualify an applicant. I strongly recommend an appointment with a competent and experienced Immigration attorney. Good luck.
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