The correction term is dr.ghs2/2, so it is the \frac{1}{2}g_{\sigma\sigma}

This term is reponsible for the stochastic steady state (no shocks, but with a precautionary motive) being different from the steady state. Your question in contrast relates to the ergodic mean (where you sample with shocks). See Simult_ and nonzero IRFs in higher-order approximations

dr.ghxu does not correspond to g_{x\sigma}. The sigma is the perturbation parameter, while the u refers to the shock. The reason there is a ghxu is that Dynare allows for a more general shock structure than the linear one in the equation below (1) in the SGU paper. In SGU, \ln A_t is part of x_t. Dynare splits this. \ln A_{t-1} is part of x_t and \varepsilon_t belongs into what Dynare calls u_t.