Hi
Having cut your timeline in two places and removed (deleted) a short section you will be left with a gap.
Right click on the gap (If it looks too narrow or you get square brackets appearing or the cursor gets in the way...zoom in a bit to make it wider.)
Now select Close Gap in Track. from the top of the menu that appears. This will remove the space and join the clips together.
Splitting and deleting the track is an intuitive way to go but if you select the clip/s on the timeline you can go into clip preview and use the red and blue START/END markers to do the trimming and the timeline is adjusted automatically....no need to delete or close a gap.
An alternative is to grab the end of the clip at the join when its on the timeline when you will see those square brackets. Grab the one you want (their shape will indicate which end you have) and simply drag it to shorten the end/start of the clip. Again the join up and changeon the timeline is automatic.
Easy for the altering the clips around the join. In the body of the clip .... Split/Split/delete/Close gap is the most logical way to do it. (There are others)
Nat