Centre of circle passing through$A\left(0,1\right),B\left(2,3\right),C\left(-2,5\right)$ is

The given points are $A\left(0,1\right),B\left(2,3\right),C\left(-2,5\right)$

Suppose that the equation of the circle is $x^2+y^2+2gx+2fy+c=0$

Substitute $A\left(0,1\right)$ in the equation 

    $1+2f+c=0\Rightarrow 2f+c=-1 ......\left(1\right)$

Substitute $B\left(2,3\right)$ in the equation

  $4+9+4g+6f+c=0\Rightarrow 4g+6f+c=-13 ......\left(2\right)$

Substitute $C\left(-2,5\right)$ in the equation

   $4+25-4g+10f+c=0\Rightarrow 4g-10f-c=29 ......\left(3\right)$

Eliminate $c$ from the equations $\left(1\right),\left(2\right),\left(3\right)$

$4g+4f=-12\Rightarrow g+f=-3 ...... \left(4\right)$

and $8g-4f=16\Rightarrow 2g-f=4 .....\left(5\right)$

Adding the above two equations, $3g=1\Rightarrow g=\frac{1}{3}$

Put$g=\frac{1}{3}$ in the equation  $g+f=-3$

hence, $f=-3-\frac{1}{3}=-\frac{10}{3}$

Therefore, the center is $\left(-g,-f\right)=\boxed{\left(-\frac{1}{3},\frac{10}{3}\right)}$