When you look at the back of the car, the right tailpipe looks much more carbonated(black) than the left side. Also when I first start the car it makes a very slight clinking noise that only lasts for about half a second. Than smoke comes out of the right tailpipe and not out of the left one. The idling is not smooth at that point and something just doesn’t feel right.
Sounds like you have an engine performance issue for sure. When is the last time you had a tune up? If the check engine light is on that means a code is stored that will help diagnose the engine related issue.
May be an engine misfire causing your idle to not be smooth. A rich or lean condition may also be a factor. Extracting any diagnostic trouble codes would be the best place to start.
Dark Smoke from the tail pipe would indicate not all fuel is being burned. If the smoke is white, this would indicate coolant being burned.