Engine sputtering and missing

The fact that this is missing after it warms up is most likely due to a sensor issue because when the engine is cold it runs on a predetermined set of parameters. Once it warms up, it starts making changes based on the sensor data specifically o2 sensors.

However, we need to limit down what sensor it could be. Is it smoking at all when it is miss firing? Is it black or or dark in color? This would indicate a rich fuel mixture.

A lean condition can also cause this and most likely there is no smoke when that is the case. Clearly this is a basic way to try and determine if it is lean or rich.

Next we need to start monitoring the sensors and see what they are telling us. The o2 sensors would be first. If they are indicating a rich or lean exhaust then we can react to that however, that should line up with if it is smoking. If they are not matching then we can suspect they are the cause. If you don't have a scan tool that can monitor data, just stop by a parts store and ask them to scan for codes because we need those as well.

Lastly the fact that you said it seems better when giving light throttle, indicates a vacuum leak.

I have attached guides for each of these items. We need to run through this and report back. Also, I attached testing that covers similar items. We need to start crossing these off the list and then we will be able to narrow down what the issue is.

https://www.2carpros.com/articles/how-to-use-an-engine-vacuum-gauge

https://www.2carpros.com/articles/engine-misfires-or-runs-rough

https://www.2carpros.com/articles/how-to-use-an-engine-vacuum-gauge