We consider a two-dimensional Hamiltonian system perturbed by a small diffusion term, whose coefficient is state-dependent and non-degenerate. As a result, the process consists of the fast motion along the level curves and slow motion across them. On finite time intervals, the large deviation principle applies, while on time scales that are inversely proportional to the size of the perturbation, the averaging principle holds, i.e., the projection of the process onto the Reeb graph converges to a Markov process. In our paper, we consider the intermediate time scales and prove the large deviation principle, with the action functional determined in terms of the averaged process on the graph.