The Saints host Newcastle on Sunday afternoon, with a win against Alan Pardew's side meaning the south coast side could move out of the bottom three in the table.
And the 23-year-old Englishman believes the board should keep faith in the 47-year-old following a late leveller against Swansea and an impressive 3-1 away win at fellow strugglers QPR.
"It would be a bit harsh if we did not give our manager a chance at this level. The last couple of results have gone quite well and, hopefully, he will get a little bit more time to show what he can do," he told reporters.
The young midfielder revealed the influence the coach has had on his career, and hopes Adkins will be rewarded for guiding the Saints to the top flight by being given time to turn things around: "I don’t know what it is like at other clubs but the manager is always there to talk to if you have a problem and he’s always got a positive attitude."
Cork has been something of a catalyst for Southampton's recent form, with the side picking up four points from six following his return from an ankle injury.
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And the youngster insists there is no time to rest for the squad as they go into another battle against Newcastle - particularly when every result could seal a manager's fate, as Mark Hughes and Roberto Di Matteo found out this week.
He continued: "There was a lot of pressure surrounding our game with QPR.
"We all knew what the consequences might have been had we lost and, unfortunately, you have seen what has happened there.
"This game can be a bit ruthless at times. Everyone is sort of fighting for their jobs every week in the Premier League.
"We must go into the Newcastle match with the same sort of attitude we've had in our last couple of games and make sure we win."