  We study AAK-type meromorphic approximants to functions $F$, where $F$ is a
sum of a rational function $R$ and a Cauchy transform of a complex measure
$\lambda$ with compact regular support included in $(-1,1)$, whose argument has
bounded variation on the support. The approximation is understood in $L^p$-norm
of the unit circle, $p\geq2$. We obtain that the counting measures of poles of
the approximants converge to the Green equilibrium distribution on the support
of $\lambda$ relative to the unit disk, that the approximants themselves
converge in capacity to $F$, and that the poles of $R$ attract at least as many
poles of the approximants as their multiplicity and not much more.
