  We present a study of two-nucleon scattering in chiral effective field theory
with a finite cutoff to next-to-leading order in the chiral expansion. In the
proposed scheme, the contributions of the lowest-order interaction to the
scattering amplitude are summed up to an arbitrary order, while the corrections
beyond leading order are iterated only once. We consider a general form of the
regulator for the leading-order potential including local and non-local
structures. The main objective of the paper is to address formal aspects of
renormalizability within the considered scheme. In particular, we provide a
rigorous proof, valid to all orders in the iterations of the leading-order
potential, that power-counting breaking terms originating from the integration
regions with momenta of the order of the cutoff can be absorbed into the
renormalization of the low energy constants of the leading contact
interactions. We also demonstrate that the cutoff dependence of the scattering
amplitude can be reduced by perturbatively subtracting the regulator artifacts
at next-to-leading order. The obtained numerical results for phase shifts in
$P$- and higher partial waves confirm the applicability of our scheme for
nucleon-nucleon scattering.
