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Choice A is correct. The x-intercept of a graph in the xy-plane is the point on the graph where $y=0$. It's given that function $h$ is defined by $h\left(x\right)=4x+28$. Therefore, the equation representing the graph of $y=h\left(x\right)$ is $y=4x+28$. Substituting $0$ for $y$ in the equation $y=4x+28$ yields $0=4x+28$. Subtracting $28$ from both sides of this equation yields $-28=4x$. Dividing both sides of this equation by $4$ yields $-7=x$. Therefore, the x-intercept of the graph of $y=h\left(x\right)$ in the xy-plane is $\left(-7,0\right)$. It's given that the x-intercept of the graph of $y=h\left(x\right)$ is $\left(a,0\right)$. Therefore, $a=-7$. The y-intercept of a graph in the xy-plane is the point on the graph where $x=0$. Substituting $0$ for $x$ in the equation $y=4x+28$ yields $y=4\left(0\right)+28$, or $y=28$. Therefore, the y-intercept of the graph of $y=h\left(x\right)$ in the xy-plane is $\left(0,28\right)$. It's given that the y-intercept of the graph of $y=h\left(x\right)$ is $\left(0,b\right)$. Therefore, $b=28$. If $a=-7$ and $b=28$, then the value of $a+b$ is $-7+28$, or $21$.

Choice B is incorrect. This is the value of $b$, not $a+b$.

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

Choice D is incorrect. This is the value of $-a+b$, not $a+b$.