We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $\beta^{(l)}$, each of which has rotated bounded variation, i.e.
\begin{equation*}
\sum_{n=0}^\infty \lvert e^{i\phi_l} \beta_{n+1}^{(l)} - \beta_n^{(l)} \rvert < \infty
\end{equation*}
for some $\phi_l$. For the real line, we impose this condition on sequences $\{a_n-1\}$ and $\{b_n\}$, where $b_n$ are the diagonal and $a_n$ the off-diagonal Jacobi coefficients, and for the unit circle, we impose it on Verblunsky coefficients. This includes discrete Schr\"odinger operators on a half-line with Wigner-von Neumann potentials $V_n = \cos(n\phi+\alpha)/n^\gamma$, with $\gamma>0$.

For the real line, our results state that in the Lebesgue decomposition $d\mu = f dm + d\mu_s$ of such measures, $\operatorname{supp}(d\mu_s) \cap (-2,2)$ is contained in a finite set $S$ (thus, there is no singular continuous part), and $f$ is continuous and non-vanishing on $(-2,2) \setminus S$. The results for the unit circle are analogous, with $(-2,2)$ replaced by the unit circle.