Background image used in “home banner widget” will look best in the default desktop view. Smaller screen size, it won’t be “really” full width (that’s the limitation of current setup of overlapping buttons).
I understand, you want to directly give home-top widget area a background image. This can be done. To extend futher, you can show a different background image for home-top to mobile users which can be of smaller width and covering 100% of smaller mobile device size.
But all this will require manual css changes.