Centre-half Clarke, fellow defender Woods, who had earlier confirmed his departure via Twitter, and midfielder Southern will not be offered new deals when their contracts expire in the summer.
Goalkeeper Ian Bennett, midfielders Chris Atkinson and Anton Robinson, plus strikers Cristian Lopez and Paul Mullin have also been released.
Out-of-contract forward Danny Ward will be offered a new deal and the club is extending defender Murray Wallace and goalkeeper Ed Wilczynski's current contracts. Striker Harry Bunn is still in talks over his future.
Former Blackpool and Southend defender Clarke, 32, joined Huddersfield in 2009 and was captain in all but two of his 224 appearances.
Woods made 81 appearances after joining on a free transfer in July 2011 from Dunfermline and Southern, who spent 10 years at Blackpool before being signed for an undisclosed fee in the summer of 2012, made 41 appearances.
He tweeted: "Really enjoyed my time at HTAFC, looking forward to the future and a fresh start.
"Just want to say a big thanks to the fans and everyone at HTAFC for your support over the last three years. Got some memories!"