Reaction-diffusion systems where transition rates exhibit quenched disorder are common in physical and chemical systems. We study pair reactions on a periodic two-dimensional lattice, including continuous deposition and spontaneous desorption of particles. Hopping and desorption are taken to be thermally activated processes. The activation energies are drawn from a binary distribution of well depths, corresponding to "shallow" and "deep" sites. This is the simplest nontrivial distribution, which we use to examine and explain fundamental features of the system. We simulate the system using kinetic Monte Carlo methods and provide a thorough understanding of our findings. We show that the combination of shallow and deep sites broadens the temperature window in which the reaction is efficient, compared to either homogeneous system. We also examine the role of spatial correlations, including systems where one type of site is arranged in a cluster or a sublattice. Finally, we show that a simple rate equation model reproduces simulation results with very good accuracy.