In run-of-the-mill criminal cases, there are two avenues of attack for incarceration: Direct appeal and a side attack using habeas corpus.
The direct appeal is limited to challenging things that happened (or should have happened) in the original trial.
Habeas is a second chance to challenge an incarceration on the basis that you are being unlawfully imprisoned (either the conditions of imprisonment - or things that weren't (and couldn't have been) part of the original trial. For example things that weren't or couldn't have been part of the original trial: if there is new evidence which could not have been discovered in the original trial, there was undiscoverable prosecutorial misconduct. Or you are being unconstitutionally imprisoned: your counsel as of right was ineffective, or the conditions in the jail/prison the original court didn't have jurisdiction.
It's a little different in immigration/deportation. In these cases, they are skipping the trial entirely and just sending folks to another jurisdiction - like El Salvador. So a habeas petition might be directed to a jurisdiction to which you had been removed in violation of a court order, or in violation of your constitutional rights to due process. So for immigration/deportation it becomes more of a direct attack because there is often nothing to appeal.
I don't think they will be able to get rid of habeas - at least not on any grounds they have cited - although habeas has been under attack (generally) since at least 1996 when Congress passed the AEDPA( Antiterrorism and Effective Death Penalty Act), which severely limited the ability to use habeas to attack death penalty convictions.
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