jmc

We could check $h(x)$ separately for each integer $x$ from $0$ to $8$: for example, $h(0)\approx 3.8$, so $h(0)>0$, but $h(1)\approx -0.7$, so $h(1)\ngtr 1$, and so on.

However, it is easier to see at a glance which $x$ satisfy $h(x)>x$ by superimposing a graph of $y=x$ on the graph of $y=h(x)$:



The six blue dots above mark the integer points $(x,x)$ which lie below the graph of $y=h(x)$, indicating that $h(x)>x$. Their $x$-coordinates are $0,3,4,5,6,8,$ which add up to $\boxed{26}$.