Given $f\left(x\right)=x^2+ax+2=0$ and $g\left(x\right)=x^2+2x+a=0$ have only one real common root, then to find the common root subtract the two equations, to get

$f\left(x\right)-g\left(x\right)=\left(a-2\right)x+2-a=0$

$\Rightarrow x=\frac{\left(a-2\right)}{\left(a-2\right)}=1$ is the common root.

Put, $x=1$ in the given equations, to get $1+a+2=0$

$\Rightarrow a=-3.$

Thus, $f\left(x\right)=x^2-3x+2=0$ and $g\left(x\right)=x^2+2x-3=0,$

$\Rightarrow f\left(x\right)+g\left(x\right)=2x^2-x-1=0.$

And, we know that the sum of roots of the quadratic equation $A{x}^2+Bx+C=0$ is $=-\frac{B}{A}.$

Thus, the sum of the roots of the equation, $f\left(x\right)+g\left(x\right)=0$ is $-\frac{\left(-1\right)}{2}=\frac{1}{2}.$