Large-scale structure projections are an obstacle in converting the shear signal of clusters detected in weak-lensing maps into virial masses. However, this step is not necessary for constraining cosmology with the shear-peak abundance, if we are able to predict its amplitude. We generate a large ensemble of N-body simulations spanning four cosmological models, with total volume V tot ~ 1(h -1 Gpc)3 per model. Variations to the matter density parameter and amplitude of fluctuations are considered. We measure the abundance of peaks in the mass density projected in ≈100 h -1 Mpc slabs to determine the impact of structures spatially correlated with the simulation clusters, identified by the three-dimensional (3D) friends-of-friends (FoF) algorithm. The halo model shows that the choice of the smoothing filter for the density field is important in reducing the contribution of correlated projections to individual halo masses. Such contributions are less than 2% in the case of the optimal, compensated filter used throughout this analysis. We measure the change in the mass of peaks when projected in slabs of various thicknesses. Peaks in slabs of 26 h -1 Mpc and 102 h -1 Mpc suffer an average mass change of less than 2% compared to their mass in slabs of 51 h -1 Mpc. We then explore the cosmology dependence of the projected-peak mass function, and find that, for a wide range of slab thicknesses (< 500 h -1 Mpc), it scales with cosmology in exactly the same way as the 3D FoF mass function and the Sheth-Tormen (ST) formula. This extends the earlier result of Marian et al. Finally, we show that for all cosmological models considered, the low and intermediate mass bins of the peak abundance can be described using a modified ST functional form to within 10%-20% accuracy.