  We study the inequality $$ {\rm div}\big(|x|^{-\alpha}|\nabla u|^{m-2}\nabla
u\big)\geq (I_\beta\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset
{\mathbb R}^N, $$ where $\alpha>0$, $N\geq 1$, $m>1$, $p, q>m-1$ and $I_\beta$
denotes the Riesz potential of order $\beta\in(0, N)$. We obtain sharp
conditions in terms of these parameters for which positive singular solutions
exist. We further establish the asymptotic profile of singular solutions to the
double inequality $$ a(I_\beta\ast u^p)u^q\geq {\rm
div}\big(|x|^{-\alpha}|\nabla u|^{m-2}\nabla u\big)\geq b(I_\beta\ast u^p)u^q
\quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $a\geq b>0$
are constants.
