As the quest for power efficient alternatives to electronics develops, magnonics has risen as a top contender. Interactions between magnons are known to massively impact the spectroscopic and transport properties at finite temperatures. In the paradigmatic antiferromagnetic honeycomb model, the authors show here that three-magnon interactions mediated by the Dzyaloshinskii-Moriya interaction increase the spin Nernst effect, whereas the four-magnon interactions mediated by exchange tend to reduce it. This competition is key to properly understanding the thermal transport of magnons at finite temperatures.