= Solution
The <posterior mean> is a <convex combination> of the <prior mean> and the usual <Poisson> <maximum-likelihood estimate>:
$$
\boxed{\mathbb E(\theta\mid\boldsymbol x)=\frac{r+s}{\lambda+n}=\frac\lambda{\lambda+n}\frac r\lambda+\frac n{\lambda+n}\bar x}.
$$
Both weights are positive and sum to one, so the mean lies between $r/\lambda$ and $\bar x$, strictly between them unless they coincide. If all counts are zero, $\bar x=0$ is the boundary estimate in the usual nonnegative Poisson parameter space; with a strictly positive parameter restriction it represents the likelihood supremum at that boundary. The displayed posterior-mean relation is valid in either interpretation.
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