Hendrie, 32, will miss the first four games of next season and has been fined £3,500 after being found guilty of an improper conduct charge.
The midfielder was charged for 'his language and behaviour' towards referee Mike Dean at the end of the Wembley final.
The Blades lost 1-0 to Burnley and Hendrie confronted the official as they left the field of play.
Bristol City have agreed new deals with midfield trio Cole Skuse, Ivan Sproule and Marvin Elliott.
Plymouth Argyle have sealed the permanent signing of midfielder