Understanding interactions between different low-energy excitations is a central theme in condensed-matter physics. Magnons and superconducting condensates represent two distinct macroscopic quantum states, yet their mutual interaction remains largely unexplored. Here, we demonstrate supercurrent-controlled magnon transport in a ferromagnet/superconductor heterostructure. We observe that the magnon transport signal decreases as the supercurrent increases. This suppression originates from a supercurrent-induced reduction of the superconducting gap, which enhances magnon dissipation into the superconductor via Bogoliubov quasiparticles. Our results provide direct experimental evidence of a supercurrent-controlled interaction between magnons and quasiparticles in the superconductor. The approach used in this study relies on supercurrents, enabling magnon modulation without Joule heating. These findings establish a new route to control magnons via superconductivity and pave the way for energy-efficient magnonic devices.