A point charge q is rotated along a circular path within the...

The electric field produced by a point charge $Q$ is an electrostatic field, which is conservative in nature.

For any conservative field, the work done by the field in moving a particle along a closed path is always zero, $\oint \vec{E} \cdot d\vec{l} = 0$.

Since one complete revolution along a circular path forms a closed loop, the total work done by the electric field on the charge $q$ is zero, irrespective of the position of the charge $Q$.