Mean-field models of neuronal populations in the brain have proven extremely useful to understand network dynamics and response to stimuli, but these models generally lack a faithful description of the fluctuations in the biologically relevant case of finite network size and connection probabilities $p<1$ (non-full connectivity). To gain insight into the mechanisms underlying the response and variability of the dynamics of populations of spiking neurons, we derive here a nonlinear stochastic mean-field model for a network of spiking Poisson neurons with quenched random connectivity. We treat the quenched disorder of the connectivity by an annealed approximation that leads to a simpler fully connected network with additional independent noise in the neurons. This annealed network enables a reduction to a low-dimensional closed system of coupled Langevin equations for the mean and variance of the neuronal membrane potentials. Compared to microscopic simulations, this mesoscopic model well describes the fluctuations and nonlinearities of finite-size neuronal populations and outperforms previous mesoscopic models that neglected the recurrent noise effect caused by quenched disorder. This effect can be analytically understood as a softening of the effective nonlinearity. The mesoscopic theory shows that quenched disorder can stabilize the asynchronous state, and it correctly predicts the large effect of connection probability and stimulus strength on the variance of the population firing rate. In conclusion, our mesoscopic theory elucidates how disordered connectivity shapes nonlinear dynamics and fluctuations of neural populations at the mesoscopic scale.