Intermediate disorder for directed polymers with space-time correlations

We revisit a result of Hairer-Shen on polymer-type approximations for the stochastic heat equation with a multiplicative noise (SHE) in $d=1$. We consider a general class of polymer models with strongly mixing environment in space and time, and we prove convergence to the It\^o solution of the SHE (modulo shear). The environment is not assumed to be Gaussian, nor is it assumed to be white-in-time. Instead of using regularity structures or paracontrolled products, we rely on simpler moment-based characterizations of the SHE to prove the convergence. However, the price to pay is that our topology of convergence is weak.