Calculation of cross-correlation of a system with noise
\begin{align}
R_{W,Y}[n] &amp;= E\big[W[m]Y[m+n]\big]\\
&amp;= E\big[W[m](Z[m+n] + W[m+n])\big]\\
&amp;= R_{W,Z}[n] + R_{W,W}[n]\\
&amp;= R_{W,Z}[n]+ P_w\delta[n].
\end{align}
So the question is: can you figure out what 
$R_{W,Z}[n] = R_{W,h\star X}[n]$ is where you know that $\{X[n]\}$ and $\{W[n]\}$ are uncorrelated processes, that is, $R_{W,X}[n] = 0$ for all $n$.
Hint: $Z[n]$ is a weighted linear combination of the $X[i]$, and, as Marcus Muller's comment points out, the curious fact about linear operations is that they are linear!