To cut the clip, place the mouse pointer over the start of the part you want to split, right-click and select 'Cut' (or press Ctrl+T). Do the same for the end of the part that you also want to cut. Now right-click the cut section and click 'Copy' (or Ctrl+C). It will now be in the computer's memory. You can now quickly rejoin that original clip by clicking on the 'Undo' button twice, or pressing Ctrl+Z twice. Now place the mouse pointer over where you want it to go on the new track, left-click the mouse, and select 'Paste', or Ctrl+V.
You guys replying to the OP need to read the question before giving answers. The OP wants to:
1. CUT the clip section 2. MOVE it to the EXACT same place on a separate track.
OP, you found a way to do it, you might want to try an alternative way:
After splitting the the clip, click on the split section to highlight it, then use the up/down arrows on the computer keyboard to move it to the new track. This is the method I use, it just seems to be easier for some reason.