I'm trying to make the cut out edges of this shape appear smooth but i can't seem to do it.
You have too many ngons, it will create artefacts, particularly on round surfaces, so what you should do is come back to your basic sphere shape, give it a Solidify + Subdivision Surface modifier, also click on Smooth Shading, cut a hole, make a circle with W > LoopTools > Circle, probably extrude, and you should be good. Apply the Solidify if you are glad with the thickness, create some additional edge loops to sharp the angle if necessary.
It looks like you have an odd geometry (ie: it has a lot of faces with different amount of edges). But there are some other things that can lead to this.
First, make sure you have auto smooth turned on. This can be found in the data section of the properties panel in the normal tab. See this post for more information.
The other thing that could be causing it has to do with the actual geometry. It's a little hard to describe and I'm not sure this will actually fix your problem but it's important to keep in mind. Basically, you can set an object to have smooth shading to make it look smoother on the surface, but that won't fix it around the edges.