In this paper, the authors propose a new second-order weak approximation scheme for hypoelliptic diffusions or degenerate stochastic differential equations satisfying a certain Hörmander condition. Moreover, they analyze the order of the variance of the proposed scheme from a mathematical aspect. Through a technique based on Malliavin calculus, they use a Gaussian process and a stochastic polynomial weight to construct the second-order weak approximation scheme, which is implemented by a Monte Carlo method and a quasi-Monte Carlo method. Numerical examples support the theoretical findings.