  Let $\Omega$ be homogeneous of degree zero, have vanishing moment of order
one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object
of investigation is the following rough non-standard singular integral operator
$$T_{\Omega,\,A}f(x)={\rm
p.\,v.}\int_{\mathbb{R}^d}\frac{\Omega(x-y)}{|x-y|^{d+1}}\big(A(x)-A(y)-\nabla
A(y)(x-y)\big)f(y){\rm d}y,$$ where $A$ is a function defined on $\mathbb{R}^d$
with derivatives of order one in ${\rm BMO}(\mathbb{R}^d)$. We show that
$T_{\Omega,\,A}$ enjoys the endpoint $L\log L$ type estimate and is $L^p$
bounded if $\Omega\in L(\log L)^{2}(\mathbb{S}^{d-1})$. These resuts
essentially improve the previous known results given by Hofmann for the $L^p$
boundedness of $T_{\Omega,\,A}$ under the condition $\Omega\in L^{q}(\mathbb
{S}^{d-1})$ $(q>1)$, Hu and Yang for the endpoint weak $L\log L$ type estimates
when $\Omega\in {\rm Lip}_{\alpha}(\mathbb{S}^{d-1})$ for some $\alpha\in
(0,\,1]$. Quantitative weighted strong and endpoint weak $L\log L$ type
inequalities are proved whenever $\Omega\in L^{\infty}(\mathbb {S}^{d-1})$. The
analysis of the weighted results relies heavily on two bilinear sparse
dominations of $T_{\Omega,\,A}$ established herein.
