The heavy-fermion compound $CeRhIn_5$ is a prototypical antiferromagnet where Ce 4f moments align antiferromagnetically below 3.8 K. When doped with Hg, the antiferromagnetic transition $T_N$ initially decreases, becomes flat, and increases again with further increasing Hg concentration. Here we report pressure effects on the electrical resistivity of a 0.45 % Hg-doped $CeRhIn_5$, where $T_N$ is 3.4 K and the magnetic structure is same as that of the undoped compound with Q=(1/2, 1/2, 0.298). With increasing pressure, $T_N$ is suppressed and a superconducting state emerges. The temperature dependence of the electrical resistivity near an optimal pressure shows a power-law behavior that deviates from a $T^2$ dependence, indicating presence of abundant quantum fluctuations near the optimal pressure.