From wikipedia:
In Bayesian inference, the conjugate prior for the rate parameter $λ$ of the Poisson distribution is the gamma distribution. Let
$$\lambda \sim \mathrm{Gamma}(\alpha, \beta)$$
denote that $λ$ is distributed according to the gamma probability density function $g$ parameterized in terms of a shape parameter $α$ and an inverse scale parameter $β$:
$$ g(\lambda \mid \alpha,\beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; \lambda^{\alpha-1} \; e^{-\beta\,\lambda} \qquad \text{ for } \lambda>0 \,$$
Then, given the same sample of $n$ measured values $k_i$, and a prior of Gamma($α$, $β$), the posterior distribution is
$$\lambda \sim \mathrm{Gamma}\left(\alpha + \sum_{i=1}^n k_i, \beta + n\right). $$
The last equation shows how to update the Gamma prior when given $n$ observations of counts $k_i$ measured from $n$ experiments.
Given my prior belief that $\lambda \sim \mathrm{Gamma}(\alpha, \beta)$, if I run another experiment to get counts of $k_1$, then I know how to update my prior to give the posterior, using $n = 1$ in the equation from wikipedia.
However, I am now told that instead of running another experiment, the count from the next experiment is a random variate from a random variable with distribution $\mathrm{Pois}(\kappa)$. Knowing this information, I wish to produce the posterior.