  We consider weighted $L^p(w)$ boundedness ($1<p<\infty $ and $w$ a
Muckenhoupt $A_p$ weight) of the Calder\'{o}n commutator $\mathcal C_\Omega$
associated with rough homogeneous kernel, under the condition $\Omega\in
L^q(\mathbb S^{n-1})$ for $q_0<q\leq\infty$ with $q_0$ a fixed constant
depending on $w$. Comparing to the previous related known results (assuming
$\Omega\in L^\infty(\mathbb S^{n-1})$), our result for $\Omega\in L^q(\mathbb
S^{n-1})$ with $q$ in the range $(q_0,\infty)$ is new. We also obtain a
quantitative weighted bound for this $\mathcal C_\Omega$ on $L^p(w)$, which is
the best known quantitative result for this class of operators.
