The line through the points (h, 3) and (4, 1) intersects the line

The slope of the line passing through points $(h, 3)$ and $(4,1)$ is

$m_{1}=\frac{1-3}{4-h}=\frac{-2}{4-h}$

The slope of line $7 x-9 y-19=0$ or $y=\frac{7}{9} x-\frac{19}{9}$ is $m_{2}=\frac{7}{9}$

It is given that the two lines are perpendicular.

$\therefore m_{1} \times m_{2}=-1$

$\Rightarrow\left(\frac{-2}{4-h}\right) \times\left(\frac{7}{9}\right)=-1$

$\Rightarrow \frac{-14}{36-9 h}=-1$

$\Rightarrow 14=36-9 h$

$\Rightarrow 9 h=36-14$

$\Rightarrow h=\frac{22}{9}$

Thus, the value of $h$ is $\frac{22}{9}$.