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Hello, with it missing, id suspect an issue with a bad injector, especially when it does it more when cold. Is the check engine light on or flashing? do you have access to a scan tool to run a diagnostic check for codes and check some live sensor data?
Ok do that and see what you is stored or pending. With a tuner, sometimes the tuner programming can become corrupt and need removed and installed, but from what you are describing it sounds like an injector has failed. P0401 is an egr fault code. Remove the egr valve and inspect for carbon buildup. With the valve removed make sure there is good flow out of where it mounts. Fire the engine up with the valve removed and check for good flow and for carbon buildup inside the intake manifold.
Remove the egr valve at the front of the engine it mounts in the intake. If the intake is plugged up with carbon buildup it will need to be removed and cleaned, if it is bad you may need to remove and clean the turbo as well. It can cause drivability issues yes, but running like it has a miss id inspect the egr first. You may still have an injector issue
Egr valve you can pull and reinstall, anything else would need replacement, intake gaskets etc. SInce you have an egr fault, need to go after that first
If you need more help just let me know, if I was helpful please leave a positive rating
Hello, do you need more help or information?