Cold hesitation problems on automatic Sportages are a common problem. It's caused by wiring issues to the MAS (Mass Airflow Sensor). It usually will NOT set a CEL.

The fix can be easy or complex, depending on the extent of the wiring problem. (Caused by corrosion and bad splices from the factory). Try the easy repair first and see if it fixes it.

The easy fix: Eliminate the electrical connector between the MAS and the engine wiring harness. This is done by cutting off the electrical connector on the harness side and soldering the wires directly to the tabs on the MAS itself. You may have to notch the plastic around the tabs to more easily get to them with a soldering iron. Make sure you note which colored wire goes to what tab on the MAS. If you mix them up you'll have real problems.

If that fixes the problem you are good to go. If it doesn't post back and I'll describe the more extensive modification that will have to be done.

BTW - The MAS is the large round unit directly after of the air filter. Has electrical connector on the back side.