Radiation generated by a charge moving in a vacuum channel through a dielectric cone is analyzed. The size of the conical target is assumed to be much greater than the wavelengths under consideration. We calculate the wave field outside the target using Stratton-Chu formulas ("aperture integrals"). This work focuses mainly on investigation of the electromagnetic field in the far-field area (Fraunhofer area). We calculate representative radiation patterns and show that typically the maximal radiation is generated in the direction of one of the refracted rays. However, in the specific case where the refracted ray is parallel to the symmetry axis, the maximal radiation is observed at some small (but nonzero) angle with respect to this axis. The maximum radiation in this case is much more intensive compared to all other situations. This phenomenon can be referred to as the "Cherenkov spotlight."