Saints came into the match under no illusions of how important it was to pick up a victory, as speculation mounted over the future of their manager. Adkins was buoyed by his players' response at St Mary's, where they looked on course for victory when Morgan Schneiderlin nodded them ahead, but a defensive lapse allowed the Swans to level.
"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."
Saturday was Cork's first Premier League appearance of the season after a frustrating run of injuries. After returning from Olympic duty with Great Britain, the 23-year-old picked up an ankle injury in a pre-season fixture at Bristol City.
Cork returned in late September but limped off 81 minutes into his comeback match against Sheffield Wednesday in the Capital One Cup with a knee injury. He has now overcome that setback and impressed on his first league game of the campaign, playing the full 90 minutes against Swansea.
"I feel really good," Cork said. "I thought I was only going to get an hour or 70 minutes, but I felt really good.
"The manager asked me at half-time how I felt and again about the hour mark and I said I felt fine. I stayed on and felt good.
"I am happy with how I played, it was just disappointing because we played really well and the lads have done great."
Source: PA
Source: PA