I am looking for a function and shortcut for it, if possible.
Lets say I have 2 clips in the timeline like the picture below. My workflow is that I play back my timeline, stop and position my playhead where I want a cut.
I cut using CTRL + K, delete the left part and ripple out the gap.
I was wondering if there was one function that would do it all, maybe one for the left side and one for the right side of a clip.
You are looking for "Ripple Trim Next Edit To Playhead" and "Ripple Trim Previous Edit To Playhead". Set them manually in the Keyboard Shortcuts dialog.
Be ware that there is a bug that will remove any transitions at the cut point in CS6 when using those commands. So it may be useful or just annoying. This is fixed in Premiere Pro CC.
Or, use Ctrl+K + select the clip you want to delete and press Shift+Delete to ripple it away in one go.
Or, set an In Point and an Out Point and use Extract to ripple away the media between then In/Out Point.