We analyze a singularly Kuramoto-Sivashinsky perturbed Camassa-Holm equation with methods of the geometric singular perturbation theory. Especially, we study the persistence of smooth and peaked solitons. Whether a solitary wave of the original Camassa-Holm equation is smooth or peaked depends on whether the parameter 2k is equal to 0, which is related to the critical wave speed. On the one hand, we prove that if 2k > 0, then a unique solitary wave persists under singular Kuramoto-Sivashinsky perturbation. On the other hand, we show that if 2k = 0, then any observable soliton fails to persist.