The edges in networks are not only binary, either present or absent, but also take weighted values in many scenarios (e.g., the number of emails between two users). The covariate-$p_0$ model has been proposed to model binary directed networks with the degree heterogeneity and covariates. However, it may cause information loss when it is applied in weighted networks. In this paper, we propose to use the Poisson distribution to model weighted directed networks, which admits the sparsity of networks, the degree heterogeneity and the homophily caused by covariates of nodes. We call it the \emph{network Poisson model}. The model contains a density parameter $μ$, a $2n$-dimensional node parameter $θ$ and a fixed dimensional regression coefficient $γ$ of covariates. Since the number of parameters increases with $n$, asymptotic theory is nonstandard. When the number $n$ of nodes goes to infinity, we establish the $\ell_\infty$-errors for the maximum likelihood estimators (MLEs), $\hatθ$ and $\hat{γ}$, which are $O_p( (\log n/n)^{1/2} )$ for $\hatθ$ and $O_p( \log n/n)$ for $\hat{γ}$, up to an additional factor. We also obtain the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings. ) for bθ and Op(log n/n) for bγ, up to an additional factor. We also obtain the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.