jmc

If $f(x)=c$ has $6$ solutions, then the horizontal line $y=c$ intersects the graph of $y=f(x)$ at $6$ points. There are two horizontal grid lines which intersect our graph $6$ times:



These lines are $y=-3,$ $y=-4$. So, the sum of all desired values of $c$ is $(-3)+(-4)=\boxed{-7}$.