—New reaction scheme is suggested for the initiated nonbranched-chain addition of hydrogen atoms to the multiple bond of the molecular oxygen. The scheme includes the addition reaction of the hydroperoxyl free radical to the oxygen molecule to form the hydrotetraoxyl free radical which is relatively low-reactive and inhibits the chain process by shortening of the kinetic chain length. This reaction competes with chain propagation reactions through a hydrogen atom. Based on the proposed scheme rate equations (containing one to three parameters to be determined directly) are deduced using quasi-steady-state treatment. The kinetic description with use the obtained rate equations is applied to the γ-induced nonbranched-chain process of the free-radical oxidation of hydrogen dissolved in water containing different amounts of oxygen at 296 K. The ratio of rate constants of competing reactions and the rate constant of the addition reaction to the molecular oxygen are defined. In this process the oxygen with the increase of its concentration begins to act as an oxidation autoinhibitor (or an antioxidant), and the rate of hydrogen peroxide formation as a function of the dissolved oxygen concentration has a maximum. From the energetic standpoint possible nonchain pathways of the free-radical oxidation of hydrogen and the routes of ozone decay via the reaction with the hydroxyl free radical (including the addition yielding the hydrotetraoxyl free radical) in the Earth’s upper atmosphere were considered.