In the present paper, we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics, viz., Riesz time-fractional Camassa–Holm (CH) equation by modified homotopy analysis method (MHAM). As a result, new types of solutions are obtained. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. The main aim of this paper is to employ a new approach, which enables us successful and efficient derivation of the analytical solutions for the Riesz time-fractional CH equation.