  Given the reproducing kernel $k$ of the Hilbert space $\mathcal{H}_k$ we
study spaces $\mathcal{H}_k(b)$ whose reproducing kernel has the form
$k(1-bb^*)$, where $b$ is a row-contraction on $\mathcal{H}_k$. In terms of
reproducing kernels this it the most far-reaching generalization of the
classical de Branges-Rovnyaks spaces, as well as their very recent
generalization to several variables. This includes the so called sub-Bergman
spaces in one or several variables. We study some general properties of
$\mathcal{H}_k(b)$ e.g. when the inclusion map into $\mathcal{H}$ is compact.
Our main result provides a model for $\mathcal{H}_k(b)$ reminiscent of the
Sz.-Nagy-Foia\c{s} model for contractions. As an application we obtain
sufficient conditions for the containment and density of the linear span of
$\{k_w:w\in\mathcal{X}\}$ in $\mathcal{H}_k(b)$. In the standard cases this
reduces to containment and density of polynomials. These methods resolve a very
recent conjecture regarding polynomial approximation in spaces with kernel
$\frac{(1-b(z)b(w)^*)^m}{(1-z\overline w)^\beta}, 1\leq m<\beta,
m\in\mathbb{N}$.
