  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 $p$-K\"ahler
structures with the $\partial\bar{\partial}$-property. Our approach is more
concerned with the $d$-closed extension by means of the exponential operator
$e^{\iota_\varphi}$. Furthermore, we prove the local stabilities of
transversely $p$-K\"ahler 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. Ra\'zny on that of the transversely K\"ahler foliations with homologically
orientability. We observe that a transversely K\"ahler foliation, even without
homologically 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 initial
foliation. Several theorems on the deformation invariance of basic
Hodge/Bott--Chern numbers with mild $\partial\bar{\partial}$-properties are
also presented.
