  In this article, we show the existence of a unique entropy solution to the
following problem: \begin{equation} \begin{split} (-\Delta)_{p,\alpha}^su&=
f(x)h(u)+g(x) ~\text{in}~\Omega,\\ u&>0~\text{in}~\Omega,\\ u&=
0~\text{in}~\mathbb{R}^N\setminus\Omega,\nonumber \end{split} \end{equation}
where the domain $\Omega\subset \mathbb{R}^N$ is bounded and contains the
origin, $ \alpha\in[0,\frac{N-ps}{2})$, $s\in (0,1)$, $2-\frac{s}{N}<p<\infty$,
$sp<N$, $g\in L^1(\Omega)$, $f\in L^q(\Omega)$ for $q>1$ and $h$ is a general
singular function with singularity at 0. Further, the fractional $p$-Laplacian
with weight $\alpha$ is given by $$(-\Delta)_{p,\alpha}^su(x)=\text{P.
V.}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\frac{dy}{|x|^\alpha|y|^{\alpha}},~\forall
x\in \mathbb{R}^N.$$
