Saints drew 1-1 with Swansea City on Saturday.
"The gaffer has done great here these two years," Cork said. "The lads have always given their all over these two years and they're not going to stop doing it now.
"Just because things aren't really going our way, the lads are always going to play the way we have played and keep playing for the gaffer.
"We know the way he likes to play and the way he likes to do it and I know the lads all like having him here. He's a good gaffer."