We consider a semilinear stochastic heat equation in spatial dimension at least 3, forced by a noise that is white in time with a covariance kernel that decays like |𝑥 | −2 as |𝑥 | → ∞. We show that in an appropriate diffusive scaling limit with a logarithmic attenuation of the noise, the pointwise statistics of the solution can be approximated by the solution to a forward-backward stochastic differential equation (FBSDE). The scaling and structure of the problem is similar to that of the two-dimensional stochastic heat equation forced by an approximation of space-time white noise considered by the first author and Gu (Ann. Probab., 2022). However the resulting FBSDE is different due to the long-range correlations of the noise.