From the context you provided, it looks like they are referring to a prison sentence. For example, if someone commits a crime, the may get a prison sentence to serve 10 years in prison. But, if their sentence is cut, perhaps they would be released from prison after only 8 years, for showing good behavior and not appearing to be a threat to anyone. It could alternately be used to refer to any punishment. For example, if a child gets into trouble, they could get a "sentence" from their parents of not being able to play their video games for one week.

"Sentence cut"= reduction of prison term/sentence