Abstract The single‐ and two‐photon blockade effects in an optical cavity with a nonlinear Kerr medium, which is coupled with a three‐level atom, forming a Δ‐type transition configuration and closed‐loop coupling are investigated by introducing two driving fields in the atom‐cavity system. It is found that single‐ and two‐photon blockades can be remarkably enhanced by the Kerr nonlinearity, however, the parameter regimes governing two blockades are distinctly different. It is demonstrated that a large Kerr nonlinearity makes more contribution to the single‐photon blockade, as compared to the two‐photon blockade effect. Furthermore, by varying the atom‐cavity coupling strength, single‐ and two‐photon blockades can alternatingly occur. Additionally, the region where two‐photon blockade occurs is widened at the cost of increased Kerr nonlinearity and atom‐cavity coupling. This work provides an alternative way to manipulate the few‐photon states and has potential applications in generating single‐photon or two‐photon sources.