Maybe. This would depend on how serious these health problems are, what stage of immigration proceedings you are in right now, and what other negative factors affect your case (for example, an existing criminal record, multiple immigration violations, etc.).
No pending proceedings in Immigration Court.
If you are not in any immigration proceedings at the moment (meaning, you are not scheduled to attend court hearings or appear in front of an Immigration Judge), you could file an application called Deferred Action (not to be confused with Deferred Action for Childhood Arrivals) with U.S. Citizenship and Immigration Services (“USCIS”). This application, if approved, will not give you a green card, but would give you a work permit that you can renew indefinitely. There is no form or formal application process for Deferred Action. This type of relief is based on discretion given by the Department of Homeland Security and a good immigration attorney can help you identify what type of evidence you will need to win a case.
Also, you need to be aware that there will be attorneys and notarios who will try to place you intentionally in removal proceedings to get you a work permit (attorney Charles Kuck previously wrote about this on the following post: http://bit.ly/1Mu7WVh). This is a very risky proposition and you should avoid it if you are eligible for Deferred Action.
Pending proceedings in Immigration Court.
If you are currently in proceedings in immigration court and have been attending court hearings, then you could seek Prosecutorial Discretion (“PD”) from the Office of Chief Counsel (the attorneys representing the government). For a description of what Prosecutorial Discretion is and how it works, please see our previous blog post here: http://bit.ly/1XzPX1t. Asking for PD is no easy task; it involves a lot of negotiating with the Office of Chief Counsel and the presentation of compelling evidence to stop your proceedings or close them.
If you do not obtain PD, you can still use evidence of your children’s serious health problems towards any applications you may have before the Court (cancellation of removal, asylum, etc.) to show that your presence is absolutely required in the U.S. and your children would suffer without you here in the country with them.
After proceedings (after an order of removal has been issued against you).
If you are already the subject of an order of removal by a Judge against you, you could apply for a stay of removal or an order of supervision with Immigration and Customs Enforcement (“ICE”), which would also give you a work permit and allow you to stay in the country indefinitely. Please take into account that if this order of removal is from January 1, 2014 or recent, then you may not be eligible for a stay of removal and may actually be detained by ICE for being a removal priority (for a discussion of what makes one a removal priority, please see our previous blog post here: http://bit.ly/1GKEGa8).
If you think you may be eligible for any of these benefits, please contact an immigration attorney to discuss the pros and cons, risks, and the different processes to obtain these benefits.
Johanna Cochran, Associate Attorney