  Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension
$n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a
general element of the linear system $|mH|$ has at worst canonical
singularities for any integer $m \geq 1$. When $X$ has terminal singularities
and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal
singularities for any integer $m \geq 1$. When $n=4$, we give an example of
Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does
not have terminal singularities.
