This inequality cannot hold in general. Indeed, at all points where $g>0$, $|\nabla g|>0$, and $|\nabla f|>0$, we can rewrite the inequality in question as 
\begin{equation}
	r-1\le C(s,q)(r^p-(p+q-2)) \tag{1}\label{1}
\end{equation}
where 
$r:=a/b$, $a:=f/g$, and $b:=|\nabla f|/|\nabla g|$. In general, $r$ can take any nonnegative real value. 

Letting now $r\to\infty$ in \eqref{1}, we get $C(s,q)>0$. Letting then $r=1$, we get a contradiction: $0\le C(s,q)(3-(p+q))<0$, since $p,q\ge2$.