The Smoluchowski-Collins-Kimball theory of irreversible diffusion-influenced reactions with one of the reactants in excess is generalized to the case of stochastic gating when one of the reactants can be in one of M states. Distinct states are characterized by various efficiencies of the reaction of contacting partners. General expressions are derived for the rate constant and for the survival probability of the reactant which is in deficiency. We present these quantities in terms of the solution of the isolated pair problem. The difference between the cases when gating is due to the reactant, which is in excess, and one, which is in deficiency, is explicitly demonstrated. General relationships between the rate constants and the survival probabilities in the two cases are established. We show. that in the former case the reaction goes faster compared to the latter one. To make the problem treatable analytically in the case when gating is due to the reactant which is in deficiency, a partial mean-field approximation is introduced. General theory is applied to, a particular case of the two-state gating model. Explicit analytical solutions for the time-dependent rate constant and the survival probability are obtained in one dimension. They illustrate the general theory was well as show how the kinetics depends on the jump rate between the two states of the gate in both cases when gating is due to the reactant, which is in excess, and one, which is in deficiency.