  We introduce a wider class of bounded Hartogs domains, which contains some
generalizations of the classical Hartogs triangle. A sharp criteria for the
$L^p-L^q$ boundedness of the Toeplitz operator with symbol $K^{-t}$ is obtained
on these domains, where $K$ is the Bergman kernel on diagonal and $t\geq 0$. It
generalizes the results by Chen and Beberok in the case $1<p<\infty$.
