I have a transparent sphere encasing a diffuse inner core with a cut out from one of its sides (shown in the attached photo). Is it possible to keep the outer material semi glossy to show reflections from the environment while hiding the annoying reflections of the inner object? Any solutions would be greatly appreciated!
Are you using Cycles or the Internal Engine to render?
I am familiar with Cycles, so I can only help you on that, assuming you're using it.
You can tell Cycles that you don't want the sphere to appear in any reflections at all, by going into the object properties, in Cycles Settings and unchecking Glossy.