The aim of this paper is to study semigroups possessing $E$-regular elements, where an element $a$ of a semigroup $S$ is {em $E$-regular} if $a$ has an inverse $a^circ$ such that $aa^circ,a^circ a$ lie in $ Esubseteq E(S)$. Where $S$ possesses `enough' (in a precisely defined way) $E$-regular elements, analogues of Green's lemmas and even of Green's theorem hold, where Green's relations $mbox{$mathcal R$},el,eh$ and $dee$ are replaced by $art_E,elt_E, eht_E$ and $widetilde{mathcal{D}}_E$. Note that $S$ itself need not be regular. We also obtain results concerning the extension of (one-sided) congruences, which we apply to (one-sided) congruences on maximal subgroups of regular semigroups.   If $S$ has an inverse subsemigroup $U$ of $E$-regular elements, such that $Esubseteq U$ and $U$ intersects every $eht_E$-class exactly once, then we say that $U$ is an {em inverse skeleton} of $S$. We give some natural examples of semigroups possessing inverse skeletons and examine a situation where we can build an inverse skeleton in a $widetilde{mathcal{D}}_E$-simple monoid. Using these techniques, we show that a reasonably wide class of $widetilde{mathcal{D}}_E$-simple monoids can be decomposed as Zappa-Sz'{e}p products. Our approach can be immediately applied to obtain corresponding results for bisimple inverse monoids.