Consider this situation: there is no cavity and no $q_C$, only a copper sphere and an external point charge $q$.
You know that in this situation the charges in the conductor rearrange so that the field in the conductor becomes zero. This is done by means of a non-uniform surface charge density $\sigma$, whose generated field in the inner of the conductor cancels with $q$'s field. This charge is located only on the surface of the sphere and its total is zero (we' re supposing a neutral conductor).
Now imagine removing a piece of the conductor so as to create the inner cavity. Will this affect the electrostatic situation? No, because the piece of conductor we are removing has zero charge density in every point, so that its presence or absence has no impact on the electric fields. The situation is the same as before, but with an inner piece of conductor replaced by empty space. Therefore, in the cavity created there is no electric field because there wasn't electric field in the corresponding inner piece of conductor.
So you can see that the external charges don't generate a field in the cavity, just like they don't generate a field in the inner of the conductor.
Finally, the effect of adding $q_C$ in the cavity can be simply figured out by means of the principle of superposition.