This paper presents 2D simulations of nanosecond discharges between two point electrodes for four different H2–air mixtures defined by their equivalence ratios phgr (i.e. phi =0, air, phi =0.3, lean mixture, phi =1, stoichiometric mixture and phi =1.5, rich mixture) at atmospheric pressure and at an initial temperature of 1000 K. In a first step, we have shown that the mixture composition has only a very small influence on the discharge dynamics and structure during the streamer phase and up to the formation of the plasma channel between the two point electrodes in H2–air mixtures with phi in [0,1.5]. However, as the plasma channel is formed slightly earlier as the equivalence ratio increases, for a given voltage pulse, the duration of the nanosecond spark phase increases as the equivalence ratio increases. As expected, we have shown that excited states of N2 (and in particular N2(A)) and radicals (and in particular O(D), O(P), H and OH) are very efficiently produced during the voltage pulse after the start of the spark phase. After the voltage pulse, and up to 100 ns, the densities of excited states of N2 and of O(D) decrease. Conversely, most of the O(P), H and OH radicals are produced after the voltage pulse due to the dissociative quenching of electronically excited N2. As for radicals, the gas temperature starts increasing after the start of the spark phase. For all studied mixtures, the density of O(P) atoms and the gas temperature reach their maxima after the end of the voltage pulse and the densities of O(P), H and OH radicals and the maximal gas temperature increase as the equivalence ratio increases. We have shown that the production of radicals is the highest on the discharge axis and the distribution of species after the voltage pulse and up to 100 ns has a larger diameter between the electrodes than close to both electrode tips. As for species, the temperature distribution presents two hot spots close to the point electrode tips. The non-uniform distributions of radical densities and gas temperature obtained after the nanosecond voltage pulse provide accurate initial conditions for 2D reactive flow codes to study the combustion ignition on longer timescales and compare with experiments.