  We consider the following model of degenerate and singular oscillatory
integral operators: \begin{equation*} Tf(x)=\int_{\mathbb{R}} e^{i\lambda
S(x,y)}K(x,y)\psi(x,y)f(y)dy, \end{equation*} where the phase functions are
homogeneous polynomials of degree $n$ and the singular kernel $K(x,y)$
satisfies suitable conditions related to a real parameter $\mu$. We show that
the sharp decay estimates on $L^2$ spaces, obtained in \cite{liu1999model}, can
be preserved on more general $L^p$ spaces with an additional condition imposed
on the singular kernel. In fact, we obtain that \begin{equation*}
\|Tf\|_{L^p}\leq C_{E,S,\psi,\mu,n,p}\lambda^{-\frac{1-\mu}{n}}\|f\|_{L^p},\ \
\frac{n-2\mu}{n-1-\mu}\leq p \leq\frac{n-2\mu}{1-\mu}. \end{equation*} The case
without the additional condition is also discussed.
