Former Celt Mackay will not resign from his post, despite seeing one of his most-trusted aides suspended, but the move has put his position under scrutiny.
Trust Chairman Tim Hartley said: "Fans want Malky Mackay, who has done a fantastic job as manager taking the club to the Premier League, to remain at Cardiff City.
"Fans deserve a clear statement from the club on what has been happening and Malky's position needs to be made perfectly clear."
Cardiff's Malaysian owner Vincent Tan has handed Moody's duties, on an acting basis, to 23-year-old Kazakhstani Alisher Apsalyamov.
Apsalyamov is thought to be a friend of Tan's son and appears to have no previous experience in football.