On the Closure of the Sum of Two Uniform Algebras on Compact Sets in $\mathbb C$

Necessary and sufficient conditions on a compact set $X$ in $\mathbb C$ and a self-homeomorphism $\psi$ of the plane $\mathbb C$ are studied under which any function continuous on $X$ can be approximated uniformly on $X$ by functions of the form $p+h\circ\psi$, where $p$ is a polynomial in a complex variable and $h$ is a rational function whose poles belong to the bounded components of the complement to the compact set $\psi(X)$.