Cherenkov radiation (CR) generated by a charge moving through a hollow conical target made of dielectric material is analyzed. We consider two cases: the charge moves from the base of the cone to its top (“straight” cone) or from the top to the base (“inverted” cone). Unlike previous papers, a nonzero shift of the charge trajectory from the symmetry axis is taken into account, which leads to generation of asymmetric CR. The most interesting effect is the phenomenon of “Cherenkov spotlight,” which was reported earlier for axially symmetric problems. This effect allows essential enhancement of CR intensity in the far-field region by proper selection of the target’s parameters and charge velocity. Here we describe the influence of charge shift on CR far-field patterns, paying most attention to the Cherenkov spotlight regime. The influence of variation of the charge speed on this phenomenon is also investigated.