(written from a Production point of view)
According to the script for Star Trek IV: The Voyage Home, Engineering Control was technical support for Federation starships in the late 23rd century. This control was located on Earth. Communication betweenn a starship and the control was coordinated by Starfleet Command.
During the Whale Probe crisis of 2286, Captain Joel Randolph of the USS Yorktown requested that Starfleet Command patch him up with the technicians at Engineering Control. He was hoping that, with their help, that they could resolve the power issues aboard his ship.