Special roots

Let the roots be $(a-d),a,(a+d)$

The sum of roots is,

$(a-d)+a+(a+d)=-\left(\dfrac{-54}{9}\right)=6$

$\begin{aligned}\Rightarrow 3a&=6\\\text{or},a&=2\end{aligned}$

now $2$ is a root of the equation thus $\hspace{2mm}f(2)=0$

$\Rightarrow 9(2^3)-54(2^2)+92(2)+k=0$

$\Rightarrow 40+k=0$

$\Rightarrow k=\boxed{-40}$