We are concerned here with unrestricted maximum likelihood estimation in a sparse $p_0$ model with covariates for directed networks. The model has a density parameter $ν$, a $2n$-dimensional node parameter $\bsη$ and a fixed dimensional regression coefficient $\bsγ$ of covariates. Previous studies focus on the restricted likelihood inference. When the number of nodes $n$ goes to infinity, we derive the $\ell_\infty$-error between the maximum likelihood estimator (MLE) $(\widehat{\bsη}, \widehat{\bsγ})$ and its true value $(\bsη, \bsγ)$. They are $O_p( (\log n/n)^{1/2} )$ for $\widehat{\bsη}$ and $O_p( \log n/n)$ for $\widehat{\bsγ}$, up to an additional factor. This explains the asymptotic bias phenomenon in the asymptotic normality of $\widehat{\bsγ}$ in \cite{Yan-Jiang-Fienberg-Leng2018}. Further, we derive the asymptotic normality of the MLE. Numerical studies and a data analysis demonstrate our theoretical findings.