How to find system function, H(z) in the z-domain, given zero-pole plot of the system?
The $z^{-1}$ term in the $H(z)$ expression you provided is the expression of the zero at infinity:

$$\lim_{z\to\infty} \frac{1}{z} = 0$$

And in your specific case:

$$\lim_{z\to\infty} H(z) = G_{0}\frac{1}{z}\frac{(z-\frac{10}{9}j)(z+\frac{10}{9}j)}{(z-\frac{9}{10})(z+\frac{9}{10})} = 0$$

Generally, if the order of the denominator exceeds the order of the numerator, then you have 1 or more zeros at $\infty$