This may seem counter-intuitive, but it seems like the best option for simply cutting a hole in a polygon (aka make a donut) would be to use any other software other than ArcGIS. Although I may be biased, and grumpy.
Shouldn’t be too hard, right? Simply start an edit session, select the area to be cut out, select cut out (in my German ArcGIS ‘ausschneiden’) and tell it which part you want to throw away. In this case, I selected the lower option, which throws away the part you just cut. Makes sense.
Double checking it….
Hm. Seems to have made a hole in the blue polygon as well. I tried to select it to delete it, but that didn’t work.
Unfortunately, it also tried to cut into my raster that was below the polygons and I ended up with this.
Moral of the story, when editing a single file like this, don’t have a single other file in your ArcGIS instance or you may end up with too many donuts. 2nd moral: make backups. I’m glad I did.