  We investigate a model-theoretic property that generalizes the classical
notion of "preservation under substructures". We call this property
\emph{preservation under substructures modulo bounded cores}, and present a
syntactic characterization via $\Sigma_2^0$ sentences for properties of
arbitrary structures definable by FO sentences. As a sharper characterization,
we further show that the count of existential quantifiers in the $\Sigma_2^0$
sentence equals the size of the smallest bounded core. We also present our
results on the sharper characterization for special fragments of FO and also
over special classes of structures. We present a (not FO-definable) class of
finite structures for which the sharper characterization fails, but for which
the classical {\L}o\'s-Tarski preservation theorem holds. As a fallout of our
studies, we obtain combinatorial proofs of the {\L}o\'s-Tarski theorem for some
of the aforementioned cases.
