It's very annoying to have to delete the cube, camera, and light every time I start a new scene. I only use Blender for games, so I never use those things. I would love for a new "General" scene to include a clean slate.
When you create a new scene, you can choose empty (new), so I guess you mean new file? So your solution should be: Delete everything, make the layout the way you want and go to File -> Defaults -> Save Startup File
Are you using 2.8 ? If it was 2.79 I would tell you to prepare your startup scene then save over the default one, or maybe after coping the old one for good measure.
Now in 2.8 we have four under
2d, Sculpting, VFX and Video Editing.
When you hit new it should ask? So in the new template system just go to app-template folder copy one of the folders, paste, rename, open it, edit and save and it should show in the start menu
If you're using 2.80, just start the scene, delete everything, and go to file - defaults - save startup file and when you start a new scene again, everything will be deleted! If you're using 2.79, save startup file will be under file.
hope this helps!