Hughes, 48, had been in charge for just short of six months after being appointed as Neale Cooper's successor in November last year.
Pools were bottom of League One when Hughes was installed and on the back of a 16-game losing streak, but the former Falkirk, Hibernian and Livingston manager was unable to save them from relegation.
A club statement read: "Hartlepool United can announce that John Hughes and Micky Barron have been given notice of termination per the relevant clauses in their contracts of employment.
"The club would like to thank John for all his efforts to try and secure League One status since he joined the club in November 2012. We wish him all the very best for the future.
"Hartlepool United would also like to thank Micky for his commitment and effort during his time as assistant, first and reserve-team coach and he also leaves with our very best wishes."
It took Hughes seven games to secure his first win in charge of the club, but a seven-game unbeaten run, that included five wins, in January gave Pools brief hope of avoiding the drop.
But they won only two of their remaining 12 games of the season and were consigned to life in League Two with three matches still to play.
Barron first joined the club as a player on loan from Middlesbrough in 1997 and went on to make 374 appearances, mainly at full-back.
He became youth-team coach in 2007 and worked as first-team coach during the 2010/11 season. Barron was twice appointed caretaker manager following the sackings of Mick Wadsworth in 2011 and Cooper in 2012.