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Inspired by a recent work of D. Wei--S. Zhu on the extension of closed complex differential forms and Voisin's usage of the $\partial\bar{\partial}$-lemma, we obtain several new theorems of deformation invariance of Hodge numbers and reprove the local stabilities of $p$-Kähler structures with the $\partial\bar{\partial}$-property. Our approach is more concerned with the $d$-closed extension by means of the exponential operator $e^{ι_φ}$. Furthermore, we prove the local stabilities of transversely $p$-Kähler structures with mild $\partial\bar{\partial}$-property by adapting the power series method to the foliated case, which strengthens the works of A. El Kacimi Alaoui--B. Gmira and P. Raźny on that of the transversely Kähler foliations with homologically orientability. We observe that a transversely Kähler foliation, even without homologically orientability, also satisfies the $\partial\bar{\partial}$-property. So even when $p=1$ (transversely Kähler), our results are new as we can drop the assumption in question on the initial foliation. Several theorems on the deformation invariance of basic Hodge/Bott--Chern numbers with mild $\partial\bar{\partial}$-properties are also presented.