  In this paper, we show that the existence of a positive weak solution to the
equation $(-\Delta_g)^s u=f u^{-q(x)}\;\mbox{in}\; \Omega,$ where $\Omega$ is a
smooth bounded domain in $R^N$, $q\in C^1(\overline{\Omega})$, and
$(-\Delta_g)^s$ is the fractional $g$-Laplacian with $g$ is the antiderivative
of a Young function and $f$ in suitable Orlicz space subjected to zero
Dirichlet condition. This includes the mixed fractional $(p,q)-$Laplacian as a
special case. The solution so obtained is also shown to be locally H\"older
continuous.
