Because of their record-low intrinsic magnetic damping properties, single-crystal yttrium-iron-garnet (YIG) films serve as an excellent model medium for studying magnon-induced spintronic phenomena such as spin pumping and the spin-orbit torque effect. For this purpose, YIG films are covered with sub-skin-depth layers of nonmagnetic heavy metals with strong spin-orbit coupling. In the present work, we show experimentally and theoretically that ohmic losses of spin-wave-induced microwave eddy currents in the heavy-metal layer deliver a strong contribution to spin-wave damping in these hybrid structures. We demonstrate that this adverse effect can be controlled and largely eliminated by placing a highly conducting metal plate near to the surface of the YIG/Pt structures. These findings are of value for a proper interpretation of experiments on the magnon spintronic effects and for the design of future magnon spintronic devices.