Wright joined the Saints in the 2012 AFL Draft, and played 35 games for the club across the past six seasons.
Freeman was acquired by the club via trade, after Collingwood picked him up with the 10th selection in the 2013 AFL Draft.
He made his debut in Round 20 this season, and played two games in the senior side at the end of the year.
Goddard played 10 games for the Saints in four seasons with the club, while Connellan failed to register a senior appearance.
“Nathan’s (Freeman) journey has obviously been well-documented, and it was a really proud moment for everyone involved in his career to see him debut this year,” St Kilda’s General Manager of Football Simon Lethlean told saints.com.au.
“He has left no stone unturned to overcome persistent hamstring issues, and his positive influence on the group despite his injuries is testament to his character.
“Hugh is another who has showed great resilience after two serious injuries, and who leaves the Saints highly respected for his professionalism and commitment to the club.
“Nathan (Wright) really set the standard at the club for selfless, team-first football, while Ray has given everything over the past two seasons to make the step from Gaelic football to the AFL.
“All four guys have made sizeable contributions to St Kilda, and we wish them all the best for the future.”