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Choice A is correct. The y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane occurs at the point on the graph where $x=0$. In other words, when $x=0$, the corresponding value of $f\left(x\right)$ is the y-coordinate of the y-intercept. Substituting $0$ for $x$ in the given equation yields $f\left(0\right)=\left(-8\right){\left(2\right)}^0+22$, which is equivalent to $f\left(0\right)=\left(-8\right)\left(1\right)+22$, or $f\left(0\right)=14$. Thus, when $x=0$, the corresponding value of $f\left(x\right)$ is $14$. Therefore, the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane is $\left(0,14\right)$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This could be the y-intercept for $f\left(x\right)=\left(-8\right){\left(2\right)}^x$, not $f\left(x\right)=\left(-8\right){\left(2\right)}^x+22$.