In this paper, the nonlinear interaction of regular water waves propagating over a fixed and submerged circular cylinder is numerically studied. At the structure’s lee side, the free surface profile experiences strong nonlinear deformation where the superharmonic free wave generated can be significant and is superposed on the transmitted wave. The wave profile then becomes asymmetric and skewed and may eventually reach the point of physical wave breaking. The governing equation and boundary conditions of this wave–structure interaction problem are formulated using both the fully nonlinear and the weak-scatterer theory. The corresponding boundary value problem is numerically solved by the immersed-boundary adaptive harmonic polynomial cell solver. In this study, a pragmatic wave-breaking suppression model is incorporated into the original solver. Both the harmonic free wave amplitudes at the structure’s lee side and the harmonic vertical forces on the cylinder are studied. The simulated harmonic wave amplitudes are compared to other published experiments and theoretical data. In general, good agreement is achieved. The effects of the incorporated wave-breaking suppression model on the simulated results are discussed. In our study, the incorporation of the pragmatic wave-breaking suppression model successfully extends the capabilities of the original fully nonlinear immersed-boundary adaptive harmonic polynomial cell solver.