Ferrimagnetic Mn$_4$N is a promising material for heat flux sensors based on the anomalous Nernst effect (ANE) because of its sizable uniaxial magnetic anisotropy ($K_{rm u}$) and low saturation magnetization ($M_{rm s}$). We experimentally and theoretically investigated the ANE and anomalous Hall effect in sputter-deposited Mn$_4$N films. It was revealed that the observed negative anomalous Hall conductivity ($sigma_{xy}$) could be explained by two different coexisting magnetic structures, that is, a dominant magnetic structure with high $K_{rm u}$ contaminated by another structure with negligible $K_{rm u}$ owing to an imperfect degree of order of nitrogen. The observed transverse thermoelectric power ($S_{rm ANE}$) of $+0.5, mu{rm V/K}$ at $300, {rm K}$ gave a transverse thermoelectric coefficient ($alpha_{xy}$) of $+0.34, {rm A/(m cdot K)}$, which was smaller than the value predicted from first-principles calculation. The interpretation for $alpha_{xy}$ based on the first-principles calculations led us to conclude that the realization of single magnetic structure with high $K_{rm u}$ and optimal adjustment of the Fermi level are promising approaches to enhance $S_{rm ANE}$ in Mn$_4$N through the sign reversal of $sigma_{xy}$ and the enlargement of $alpha_{xy}$ up to a theoretical value of $1.77, {rm A/(m cdot K)}$.
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