The halo mass function (HMF) is a critical element in cosmological analyses of galaxy cluster catalogs. We quantify the impact of uncertainties in HMF parameters on cosmological constraints from cluster catalogs similar to those from Planck, those expected from the Euclid, Roman, and Rubin surveys, and from a hypothetical larger future survey. We analyze simulated catalogs in each case, gradually loosening priors on HMF parameters to evaluate the degradation in cosmological constraints. While current uncertainties on HMF parameters do not substantially impact Planck-like surveys, we find that they can significantly degrade the cosmological constraints for a Euclid-like survey. Consequently, the current precision on the HMF is not sufficient for Euclid, Roman, Rubin or possible larger surveys. Future experiments will have to properly account for uncertainties in HMF parameters, and it will be necessary to improve the precision of HMF fits to avoid weakening constraints on cosmological parameters.Key words: galaxies: clusters: general / large-scale structure of Universe / cosmological parameters