If you have DC current present in a transformer with a ferro-magnetic core then it will bias the flux in the core so the core does not saturate symmetrically, thus reducing the maximum alternating flux which can be tolerated. If you increase the DC current further the core will saturate and the relationship between primary and secondary windings will become highly non-linear.Introducing an air gap into the path of the flux improves ability of the core to tolerate DC current, at the expense of degrading other core parameters.