The neutron star (NS) mass distribution serves as a crucial probe of the dense matter equation of state (EOS) and core-collapse supernova explosion mechanisms. Although observations reveal a bimodal NS mass distribution, existing theoretical simulations often struggle to naturally reproduce the high-mass peak, and the impact of the quark-hadron phase transition (PT) in high-density nuclear matter remains largely unexplored. Employing the 1D general-relativistic hydrodynamics code GR1D with hybrid EOSs incorporating a first-order PT (RDF1.1 and RDF1.2), we systematically simulate core-collapse supernovae for progenitors with zero-age main-sequence masses ranging from 13 to 120 M⊙. Our results reveal a threefold impact of the PT onset density on the final supernova fate: in low-compactness progenitors, an early PT truncates accretion and produces less massive NSs; in the critical-compactness regime, it facilitates successful explosions; and in high-compactness progenitors, it accelerates black hole formation. Compared with the ∼ 1.4 M⊙ peak typical of the conventional core-collapse scenario, the inclusion of the PT produces NSs with masses centered around 1.70 M⊙ for RDF1.1 and within 1.55–1.70 M⊙ for RDF1.2. Furthermore, we identify a novel formation channel whereby low-compactness progenitors with failed initial shocks can form more massive NSs via a PT induced second shock. Due to the fixed nature of the PT boundary, the final NS masses exhibit a pronounced clustering effect. This study elucidates the critical role of the quark-hadron PT in shaping the NS mass distribution, providing important astrophysical constraints on the properties of high-density nuclear matter and the formation mechanism of massive NSs.