  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 $1.77\, {\rm A/(m \cdot K)}$.
