= Solution
Under the intended assumption that the two white-noise sequences are mutually uncorrelated at every pair of times, $X$ and $W$ are uncorrelated. Their sum is therefore weakly stationary with
$$
\mathbb EY_t=0,
\qquad
\gamma_Y(h)=
\frac{\sigma^2}{1-\phi^2}\phi^{|h|}
+\sigma_W^2\mathbf1_{\{h=0\}}.
$$
Strictly, the printed condition $\mathbb E[\varepsilon_tW_t]=0$ only at equal times is insufficient. For example, $W_t=(-1)^t\varepsilon_{t-1}$ is itself white noise and is contemporaneously uncorrelated with $\varepsilon_t$, but the cross-covariance contribution can depend on $t$. The displayed answer therefore uses the standard intended cross-series white-noise assumption $\mathbb E[\varepsilon_tW_s]=0$ for all $s,t$.
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