Efficient manipulation of magnon spin transport is crucial for developing magnon-based spintronic devices. In this Letter, we provide proof of principle of a method for modulating the diffusive transport of thermal magnons in an yttrium iron garnet channel between injector and detector contacts. The magnon spin conductance of the channel is altered by increasing or decreasing the magnon chemical potential via spin Hall injection of magnons by a third modulator electrode. We obtain a modulation efficiency of 1.6%/mA at T=250 K. Finite element modeling shows that this could be increased to well above 10%/mA by reducing the thickness of the channel, providing interesting prospects for the development of thermal-magnon-based logic circuits.