We characterize those $\alpha\in \mathbb{R}$ and $\mu$ positive Borel measure on $(0,1]$ for which generalized Hausdorff operator acts on Hardy spaces of the unit disk. Further, certain conditions on $\mu,$ we prove the operator is bounded linear on $H^p(\mathbb{D}),$ for different cases of $p.$ For $\alpha=0,$ we determine the characterization of the operator on weighted spaces of integrable functions.