Let $$f = P[F]$$ denote the Poisson integral of F in the unit disk $${\mathbb {D}}$$ with F being absolutely 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 $$p\ge 1$$ . Recently, the author in Zhu (J Geom Anal, 2020) proved that (1) if f is a harmonic mapping and $$1\le p< 2$$ , then $$f_{z}$$ and $$\overline{f_{{\overline{z}}}}\in \mathcal {B}^{p}({\mathbb {D}}),$$ the classical Bergman spaces of $${\mathbb {D}}$$ [12, Theorem 1.2]; (2) if f is a harmonic quasiregular mapping and $$1\le p\le \infty $$ , then $$f_{z},$$ $$\overline{f_{{\overline{z}}}}\in \mathcal {H}^{p}({\mathbb {D}}),$$ the classical Hardy spaces of $${\mathbb {D}}$$ [12, Theorem 1.3]. These are the main results in Zhu (J Geom Anal, 2020). The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, [12, Theorem 1.2] is true when $$1\le p< \infty $$ . Also, we show that [12, Theorem 1.2] is not true when $$p=\infty $$ . Second, we demonstrate that [12, Theorem 1.3] still holds true when the assumption f being a harmonic quasiregular mapping is replaced by the weaker one f being a harmonic elliptic mapping.