Hi,
No, there's no permanent 'cut' feature but there is a way to achieve what you want:
1. Right-click (or Ctrl+click on the Macintosh) in the list of Recordings / imported files in the Cleanup Audio window and select 'Save a Copy' ... 'In VinylStudio's Recordings Folder'.
2. In the confirmation dialog, select 'Save corrected audio'.
3. Save the file.
You can then delete your original recording and the copy will be shorter by the length of the cut(s).
Having said which, I wouldn't bother unless the cuts are long. You probably won't save much and you lose the ability to undo any other changes you have made to the original file.
VinylStudio saves your work on shutdown and every 5 minutes by default. You can change this in the Settings / Preferences dialog.