A semigroup
$S$
is called idempotent-surjective (respectively, regular-surjective) if whenever
$\rho $
is a congruence on
$S$
and
$a\rho $
is idempotent (respectively, regular) in
$S/ \rho $
, then there is
$e\in {E}_{S} \cap a\rho $
(respectively,
$r\in \mathrm{Reg} (S)\cap a\rho $
), where
${E}_{S} $
(respectively,
$\mathrm{Reg} (S)$
) denotes the set of all idempotents (respectively, regular elements) of
$S$
. Moreover, a semigroup
$S$
is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.