Cortese's departure sparked talk of meltdown at St Mary's but Saints showed their mettle on the pitch with a 2-2 draw at Sunderland on Saturday.
Pochettino's side sit in ninth in the Premier League table - well clear of relegation trouble - and Cork believes the club must keep hold of Pochettino next term to continue to push on.
While the Argentine tactician said last May it would "make no sense" to remain at Southampton without Cortese, he has since pledged "100 per cent" to see out the current campaign at St Mary's.
Cork said: "For footballers it is the manager who matters, he is the one who deals with us.
"He is the one we see every day, talks to us and gives us the confidence to play. What happens above that should not affect us.
"How vital is it he stays next season too? Very. He has been a great manager for us. It is vital he stays and we keep this thing going next season.
"As far as we are concerned, it is as you were. We were laughing at talk of us imploding. That was never going to happen. There has been a lot of silly talk. The gaffer came in and said I am not going anywhere. We carry on. That was nice to hear.
"I don't think it affected the lads much, but it was good for him to come in and say that.
"Sometimes when you are in a bad environment, when the finger of blame is out, you don't want to play the football. That can make it hard to play.
"But funnily enough it has not been much different to other weeks, even though there was all that talk of meltdown and such. It has never felt like that. We go in and train and play football like we always do."