  A Hermitian and an anti-Hermitian first-order intertwining operators are
introduced and a class of $\eta$-weak-pseudo-Hermitian position-dependent mass
(PDM) Hamiltonians are constructed. A corresponding reference-target
$\eta$-weak-pseudo-Hermitian PDM -- Hamiltonians' map is suggested. Some
$\eta$-weak-pseudo-Hermitian PT -symmetric Scarf II and periodic-type models
are used as illustrative examples. Energy-levels crossing and flown-away states
phenomena are reported for the resulting Scarf II spectrum. Some of the
corresponding $\eta$-weak-pseudo-Hermitian Scarf II- and
periodic-type-isospectral models (PT -symmetric and non-PT -symmetric) are
given as products of the reference-target map.
