There is not enough information in your question. You should consult with an immigration attorney about the situation.
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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.
This is general information and not legal advise. This communication does not create a lawyer-client relationship.
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.
This is general information, not legal advice, and does not create an attorney client relationship.
Many different ways but none through a direct immigration petition.
NYC EXPERIENCED IMMIGRATION ATTORNEYS www.myattorneyusa.com; email: firstname.lastname@example.org; Phone: (866) 456-8654; Fax: 212-964-0440; Cell: 212-202-0325. The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.