  Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk
$\mathbb{D}$ with $F$ is an absolute continuous in the unit circle $\mathbb{T}$
and $\dot{F}\in L^p(\mathbb{T})$, where $\dot{F}(e^{it}) = \frac{d}{dt}
F(e^{it})$ and $p \in [1,\infty]$. Recently, Chen et al. (J. Geom. Anal., 2021)
extended Zhu's results (J. Geom. Anal., 2020) and proved that (i) if $f$ is a
harmonic mapping and $1 \leq p < \infty$, then $f_z$ and
$\overline{f_{\overline{z}}} \in B^p(\mathbb{D})$, the Bergman spaces of
$\mathbb{D}$. Moreover, (ii) under additional conditions as $f$ being harmonic
quasiregular mapping in \cite{Zhu} or $f$ being harmonic elliptic mapping in
\cite{CPW}, they proved that $f_z$ and $\overline{f_{\overline{z}}}\in
H^p(\mathbb{D})$, the Hardy space of $\mathbb{D}$, for $1 \leq p \leq \infty$.
The aim of this paper is to extend these results by showing that (ii) holds for
$p\in(1,\infty)$ without any extra conditions and for $p=1$ or $p=\infty$,
$f_z$ and $\overline{f_{\bar{z}}}\in H^p(\mathbb{D})$ if and only if
$H(\dot{F})\in L^p(\mathbb{T})$, the Hilbert transform of $\dot{F}$ and in that
case, it yields $zf_z=P[\frac{\dot{F}+iH(\dot{F})}{2i}]$.
