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Inspired by a recent work of D. Wei--S. Zhu on the extension of closedcomplex differential forms and Voisin's usage of the$\partial\bar{\partial}$-lemma, we obtain several new theorems of deformationinvariance of Hodge numbers and reprove the local stabilities of $p$-K\"ahlerstructures with the $\partial\bar{\partial}$-property. Our approach is moreconcerned with the $d$-closed extension by means of the exponential operator$e^{\iota_\varphi}$. Furthermore, we prove the local stabilities oftransversely $p$-K\"ahler structures with mild$\partial\bar{\partial}$-property by adapting the power series method to thefoliated case, which strengthens the works of A. El Kacimi Alaoui--B. Gmira andP. Ra\'zny on that of the transversely K\"ahler foliations with homologicallyorientability. We observe that a transversely K\"ahler foliation, even withouthomologically orientability, also satisfies the$\partial\bar{\partial}$-property. So even when $p=1$ (transversely K\"ahler),our results are new as we can drop the assumption in question on the initialfoliation. Several theorems on the deformation invariance of basicHodge/Bott--Chern numbers with mild $\partial\bar{\partial}$-properties arealso presented.

On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-K\"ahler structures