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Choice B is correct. It's given that the graph of $y=g\left(x\right)$ is produced by translating the graph of $y=f\left(x\right)$ $3$ units down and $4$ units to the right in the xy-plane. Therefore, function $g$ can be defined by an equation in the form $g\left(x\right)=f\left(x-4\right)-3$. Function $f$ is defined by the equation $f\left(x\right)=\frac{a-19}{x}+5$, where $a$ is a constant. Substituting $x-4$ for $x$ in the equation $f\left(x\right)=\frac{a-19}{x}+5$ yields $f\left(x-4\right)=\frac{a-19}{x-4}+5$. Substituting $\frac{a-19}{x-4}+5$ for $f\left(x-4\right)$ in the equation $g\left(x\right)=f\left(x-4\right)-3$ yields $g\left(x\right)=\frac{a-19}{x-4}+5-3$, or $g\left(x\right)=\frac{a-19}{x-4}+2$. Therefore, the equation that defines function $g$ is $g\left(x\right)=\frac{a-19}{x-4}+2$.

Choice A is incorrect. This equation defines a function whose graph is produced by translating the graph of $y=f\left(x\right)$ $3$ units down and $4$ units to the left, not $3$ units down and $4$ units to the right.

Choice C is incorrect. This equation defines a function whose graph is produced by translating the graph of  $y=f\left(x\right)$ $4$ units to the left, not $3$ units down and $4$ units to the right.

Choice D is incorrect. This equation defines a function whose graph is produced by translating the graph of $y=f\left(x\right)$ $4$ units to the right, not $3$ units down and $4$ units to the right.