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Choice D is correct. It's given that a wire with a length of $106$ inches is cut into two parts. It's also given that one part has a length of $x$ inches and the other part has a length of $y$ inches. This can be represented by the equation $x+y=106$. It's also given that the value of $x$ is $6$ more than $4$ times the value of $y$. This can be represented by the equation $x=4y+6$. Substituting $4y+6$ for $x$ in the equation $x+y=106$ yields $4y+6+y=106$, or $5y+6=106$. Subtracting $6$ from each side of this equation yields $5y=100$. Dividing each side of this equation by $5$ yields $y=20$. Substituting $20$ for $y$ in the equation $x=4y+6$ yields $x=4\left(20\right)+6$, or $x=86$. 

Choice A is incorrect. This value represents less than half of the total length of $106$ inches; however, $x$ represents the length of the longer part of the wire, since it's given that the value of $x$ is $6$ more than $4$ times the value of $y$.

Choice B is incorrect. This value represents less than half of the total length of $106$ inches; however, $x$ represents the length of the longer part of the wire, since it's given that the value of $x$ is $6$ more than $4$ times the value of $y$.

Choice C is incorrect. This represents a part that is $6$ more than the length of the other part, rather than $6$ more than $4$ times the length of the other part.