  This article consists of two connected parts. In the first part, we study the
shift invariant subspaces in certain $\mathcal{P}^2(\mu)$-spaces, which are the
closures of analytic polynomials in the Lebesgue spaces $\mathcal{L}^2(\mu)$
defined by a class of measures $\mu$ living on the closed unit disk
$\overline{\mathbb{D}}$. The measures $\mu$ which occur in our study have a
part on the open disk $\mathbb{D}$ which is radial and decreases at least
exponentially fast near the boundary. Our focus is on those shift invariant
subspaces which are generated by a bounded function in $H^\infty$. In this
context, our results are definitive. We give a characterization of the cyclic
singular inner functions by an explicit and readily verifiable condition, and
we establish certain permanence properties of non-cyclic ones which are
important in the applications. The applications take up the second part of the
article. We prove that if a function $g \in \mathcal{L}^1(\mathbb{T})$ on the
unit circle $\mathbb{T}$ has a Cauchy transform with Taylor coefficients of
order $\mathcal{O}\big(\exp(-c \sqrt{n})\big)$ for some $c > 0$, then the set
$U = \{x \in \mathbb{T} : |g(x)| > 0 \}$ is essentially open and $\log |g|$ is
locally integrable on $U$. We establish also a simple characterization of
analytic functions $b: \mathbb{D} \to \mathbb{D}$ with the property that the de
Branges-Rovnyak space $\mathcal{H}(b)$ contains a dense subset of functions
which, in a sense, just barely fail to have an analytic continuation to a disk
of radius larger than 1. We indicate how close our results are to being optimal
and pose a few questions.
