Suppose $$p\ge 1$$ , $$w=P[F]$$ is a harmonic mapping of the unit disk $${\mathbb D}$$ satisfying F is absolutely continuous and $${\dot{F}}\in L^p(0, 2\pi )$$ , where $${\dot{F}}(e^{it})=\frac{\mathrm {d}}{\mathrm {d}t}F(e^{it})$$ . In this paper, we obtain Bergman norm estimates of the partial derivatives for w, i.e., $$\Vert w_z\Vert _{L^p}$$ and $$\Vert \overline{w_{{\bar{z}}}}\Vert _{L^p}$$ , where $$1\le p<2$$ . Furthermore, if w is a harmonic quasiregular mapping of $${\mathbb {D}}$$ , then we show that $$w_z$$ and $$\overline{w_{{\bar{z}}}}$$ are in the Hardy space $$H^p$$ , where $$1\le p\le \infty $$ . The corresponding Hardy norm estimates, $$\Vert w_z\Vert _{p}$$ and $$\Vert \overline{w_{{\bar{z}}}}\Vert _{p}$$ , are also obtained.