  Let $M(\phi)=T(\phi)+H(\phi)$ be the Toeplitz plus Hankel operator acting on
$H^p(\T)$ with generating function $\phi\in L^\iy(\T)$. In a previous paper we
proved that $M(\phi)$ is invertible if and only if $\phi$ admits a
factorization $\phi(t)=\phi_{-}(t)\phi_{0}(t)$ such that $\phi_{-}$ and
$\phi_{0}$ and their inverses belong to certain function spaces and such that a
further condition formulated in terms of $\phi_{-}$ and $\phi_{0}$ is
satisfied. In this paper we prove that this additional condition is equivalent
to the Hunt-Muckenhoupt-Wheeden condition (or, $A_{p}$-condition) for a certain
function $\sigma$ defined on $[-1,1]$, which is given in terms of $\phi_{0}$.
As an application, a necessary and sufficient criteria for the invertibility of
$M(\phi)$ with piecewise continuous functions $\phi$ is proved directly.
Fredholm criteria are obtained as well.
