<p>The growing energy consumption of the computational sector worldwide necessitates the search for sustainable methods of powering and performing calculations. The fast-emerging field of spin caloritronics offers a promising solution by combining the advantages of performing computations on spins instead of charges and driving these computations through temperature differences rather than voltage. Among the various approaches, spin waves and their quanta of excitations, known as magnons, are considered promising carriers of spin-encoded information. In this article, we examine the magnon current generated in a system composed of a magnetic insulator/quantum dot/magnetic insulator, driven by a small temperature difference applied between the two insulators. By expanding the magnon current in terms of the applied temperature bias, we analyze the contributions of successive terms up to the third order of the temperature difference. Each term exhibits a similar structure, consisting of a driving-like and a damping-like component. The driving-like term is dependent on the coupling strength between the quantum dot and the electrodes. We explicitly show that the second-order term of the magnon current vanishes when the couplings of the quantum dot to the magnetic insulators are equal. Overall, the first two terms are sufficient to capture the behavior of the magnon current across the full range of temperature differences. For extreme values of the temperature gradient, the approximate results align with the exact ones only when there is significant asymmetry in the coupling strengths. Finally, we demonstrate that the system can function as a spin diode, capable of rectifying the magnon current when the temperature bias is reversed.</p>