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Choice D is correct. The given phrase “$8$ times a number $x$” can be represented by the expression $8x$. The given phrase “$3$ more than” indicates an increase of $3$ to a quantity. Therefore “$3$ more than $8$ times a number $x$” can be represented by the expression $8x+3$. Since it’s given that $3$ more than $8$ times a number $x$ is equal to $83$, it follows that $8x+3$ is equal to $83$, or $8x+3=83$. Therefore, the equation that represents this situation is $8x+3=83$.

Choice A is incorrect. This equation represents $3$ times the quantity $8$ times a number $x$ is equal to $83$. 

Choice B is incorrect. This equation represents $8$ times a number $x$ is equal to $3$ more than $83$.

Choice C is incorrect. This equation represents $8$ more than $3$ times a number $x$ is equal to $83$.