= Solution
The new <prior density> contributes the factor $\lambda_i^{15}e^{-16\lambda_i}$. Multiplication by the <Poisson distribution> <likelihood> gives
$$
\boxed{\lambda_i\mid y_i\sim\operatorname{Gamma}(y_i+16,E_i+16).}
$$
Its <posterior mean> displays the resulting shrinkage:
$$
\mathbb E(\lambda_i\mid y_i)=\frac{y_i+16}{E_i+16}
=\frac{E_i}{E_i+16}\frac{y_i}{E_i}+\frac{16}{E_i+16}\cdot1.
$$
The observed risk ratio $y_i/E_i$ is pulled towards one, most strongly for small exposures. The <posterior variance> is $(y_i+16)/(E_i+16)^2$.
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