jmc

We note that $$q(x)=-|x|=\{\begin{array}{ll}x& \text{if }x\le 0\\ -x& \text{if }x>0\end{array}.$$Therefore, $$q(p(x))=-|p(x)|=\{\begin{array}{ll}p(x)& \text{if }p(x)\le 0\\ -p(x)& \text{if }p(x)>0\end{array}.$$A graph of $y=q(p(x))$ looks like the graph of $y=p(x)$ with the parts above the $x$-axis reflected so that they lie below the $x$-axis: The graph readily shows us that the values of $q(p(x))$ at $x=-4,$ $-3,$ $-2,$ $-1,$ $0,$ $1,$ $2,$ $3,$ $4$ are respectively $-2,$ $-1,$ $0,$ $-1,$ $-2,$ $-1,$ $0,$ $-1,$ $-2.$ The sum of these values is $\boxed{-10}.$