Abstract: Formation of transverse modes in a dual-polarization self-mode-locked monolithic Yb: KGW laser under high-power pumping is thoroughly explored. It is experimentally observed that the polarization-resolved transverse patterns are considerably affected by the pump location in the transverse plane of the gain medium. In contrast, the longitudinal self-mode-locking is nearly undisturbed by the pump position, even under the high-power pumping. Under central pumping, a vortex beam of the Laguerre-Gaussian LGp,l mode with p = 1 and l = 1 can be efficiently generated through the process of the gain competition with a sub-picosecond pulse train at 25.3 GHz and the output power can be up to 1.45 W at a pump power of 10.0 W. Under off-center pumping, the symmetry breaking causes the transverse patterns to be dominated by the high-order Hermite-Gaussian modes. Numerical analyses are further performed to manifest the symmetry breaking induced by the off-center pumping.