Earlier this week the men received letters notifying them their names had been removed from the list of tens of thousands of people prohibited from flying to, from or within the United States.
Three of the four Jameel Albighah, Naveed Shinwari and Awais Sajjad were in federal court in Manhattan to hear the arguments over a government motion to have their lawsuit against 25 FBI agents dismissed.
As a result, they were unable to travel to see loved ones, lost job opportunities and suffered the "stigma of being treated as threats to aviation security," he said.
He added: "Money relief is really the only relief."
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In a statement, Albighah, a US citizen from Yemen with a wife and three daughters who still live there, said being on the list proved agonizing.
"They messed up my life," he said. "My youngest daughter doesn't even know me."
Abrams is expected to rule later in a written decision. In April, the Obama administration announced a new policy allows for an American traveler who has been denied boarding a commercial airliner to petition the US Transportation Security Administration once to find out whether he or she is on the no-fly list, and for an unclassified explanation of why he or she is on the list.
But the American Civil Liberties Union, which has been challenging the constitutionality of the no-fly list in a separate federal lawsuit, have said the changes don't go far enough in giving travelers the legal due process they are entitled to, including seeing the evidence held against them and getting an opportunity to challenge it.