The Tykes were relegated from the Championship on Friday and the departure of head coach Poya Asbaghi followed on Sunday, with Martin Devaney put in charge for the final three games of the campaign.
Manchester City youngster Gomes arrived at the end of the summer window while Moroccan playmaker Bassi joined the club in the January.
Exciting forward Quina also made the short-term move from Watford in the winter window.
Bassi made 15 appearances, scoring two goals and providing five assists, while Quina played 18 times and scored twice. Gomes made 31 appearances for the Tykes, scoring once and assisting once from defensive midfield.
Chief Executive Officer, Khaled El-Ahmad: “I would like to thank all of the players for their commitment and effort. Each individual has played a part this season and all at Oakwell would like to wish them well for the future.”