The estimation of sums of functions of observable and unobservable variables is a long-standing problem in statistics, with applications in many domains. We consider this problem in Poisson mixture models, where empirical Bayes provides a natural framework but nonparametric theory remains limited. We develop a nonparametric empirical Bayes methodology based on regular and coarsened minimum-distance estimation of the unknown mixing distribution. For a broad class of such sums, we establish large-sample guarantees showing that the resulting plug-in estimates asymptotically merge with the oracle Bayes estimate. In particular, when the mixing distribution has finite support, we obtain a nearly parametric convergence rate, up to a logarithmic factor. We then provide a finite-sample analysis of two representative sums: the total intensity among units whose observed count does not exceed a fixed threshold, and the number of units whose observed count exceeds their latent intensity. For the total-intensity sum, we establish a lower bound on the minimax regret and derive upper bounds for both regular and coarsened minimum-distance procedures under compact-support and subexponential assumptions on the mixing distribution. These upper bounds match the minimax rate up to logarithmic factors, with the coarsened procedure in the compact-support setting matching the rate exactly when the coarsening level is fixed. The above-average sum displays a markedly different behavior: a lower bound on the minimax regret shows that bounded regret is in general impossible to achieve. We identify additional stability conditions under which bounded regret can be recovered up to logarithmic factors, and show that these conditions are automatically satisfied when the mixing distribution has finite support. Numerical experiments on both synthetic and real data illustrate the performance of the proposed methodology.

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