  We derive normal approximation results for a class of stabilizing functionals
of binomial or Poisson point process, that are not necessarily expressible as
sums of certain score functions. Our approach is based on a flexible notion of
the add-one cost operator, which helps one to deal with the second-order cost
operator via suitably appropriate first-order operators. We combine this
flexible notion with the theory of strong stabilization to establish our
results. We illustrate the applicability of our results by establishing normal
approximation results for certain geometric and topological statistics arising
frequently in practice. Several existing results also emerge as special cases
of our approach.
