  We study the one-dimensional non-Hermitian lattices with staggered onsite
modulations and nonreciprocal hopping up to the next-nearest-neighboring (NNN)
sites. Due to the NNN nonreciprocity, the non-Hermitian skin effect (NHSE) in
the system under open boundary conditions (OBC) can be energy dependent, and
there will be NHSE edges in the eigenenergy spectrum, which separates the
eigenstates localized at the opposite ends of the lattice. We find that the
interplay between the nonreciprocal hopping and onsite modulations can reverse
the direction of the skin effect and modify the position of the NHSE edge.
Moreover, by tuning the system parameters, some of the eigenstates under OBC
will become fully extended with the corresponding eigenenergies being imaginary
under both open and periodic boundary conditions. Thus, the extended states can
coexist with the NHSH in the same system. The NHSE can even be completely
dissolved with all the eigenstates being extended when the modulation is
imaginary. Our work unveils the intricate interplay between onsite modulations
and nonreciprocal hopping in non-Hermitian systems.
