Circle touches the $x$ -axis at $\left(1,\mathrm{}0\right)$
So, center of circle is $\left(1,\mathrm{}k\right)$ and radius is $k.$
Hence, the equation of circle is:
${\left(x-1\right)}^2+{\left(y-k\right)}^2=k^2$
It passes through the point $\left(2,\mathrm{}3\right)$ ,

$\Rightarrow 1+9-6k+k^2=k^2$

$\therefore k=\frac{5}{3}$