In the past, Cubans were generally allowed to live and work in Florida without much trouble. However, that policy is starting to change, and it has resulted in Cuban nationals being deported or not allowed to enter the United States at all. One man was taken into custody upon returning to Miami after his honeymoon. He was a permanent resident and had been in the United States since he was a child.
After a court hearing, he was granted a hardship waiver that allowed him to remain in the United States. The change in treatment of Cuban citizens began when Barack Obama was president and the "wet-foot, dry-foot" policy was put to rest. It had allowed those from Cuba to remain in the United States if they managed to reach dry land. However, only those who had committed violent crimes were made a priority to be deported back to Cuba.
President Trump has made it a priority to deport anyone believed to be in the country illegally. Furthermore, he announced that visas for Cuban citizens would only last for three months. To apply for a green card, a Cuban national has to be in the United States legally for a year and a day. This is according to the Cuban Adjustment Act signed into law in 1966.
Those who are facing removal proceedings will not necessarily be deported back to their country of origin. It may be possible to apply for asylum, ask for a hardship waiver or dispute the facts of the case in court. An attorney may be able to help a person establish that he or she has the right to remain in the country.
Source: NPR, "Cuban Immigrants Were Given A Haven In The U.S.; Now They're Being Deported", Daniel Rivero, May 11, 2019