The Scot's future was in doubt this week when a trusted aid of his was sacked and replaced by a 23-year-old novice.
After a five-hour meeting earlier this week, Mackay has remained in charge and promised the fans that recent issues will not affect the team on Saturday.
"I do want to apologise to our supporters because I understand completely what they are going through," Mackay told reporters.
"I think that is needed. In the two-and-a-half years I have been here I feel I have forged a big bond with the fans and the people of Cardiff, they have been fantastic to me.
"I do regret that their club has been thrown into the limelight for reasons other than our football. Our preparations this week, apart from the fact we had an international break, have been exactly as they always are."
Cardiff take on Chelsea at 3pm on Saturday.