nature of output of the addition of white noise with colord noise
Are $x$ and $y$ uncorrelated? In this case the autocorrelation function of $x+y$ is just the sum of the autocorrelation functions of $x$ and $y$, which if one of them is not white will lead to its sum also not being white.
In general, for stationary processes you have $\phi_{zz}(\tau) = \phi_{xx}(\tau) + \phi_{yy}(\tau) + \phi_{xy}(\tau) + \phi_{yx}(\tau)$ where $\phi_{xy}(\tau) = \mathbb{E}\{x(t)y(t+\tau)\}$. Moreover, since the process is stationary you also have $\phi_{xy}(\tau) = \phi_{xy}(-\tau) = \phi_{yx}(\tau)$. So in general, the autocorrelation of $z$ depends not only on the autocorrelations of $x$ and $y$ but also their cross correlations.