We consider a dynamic system that is driven by an intensity-modulated Poisson process with intensity Λ(t) = λ(t)+εν(t). We derive an exact relation between the input-output cross-correlation in the spontaneous state (ε = 0) and the linear response to the modulation (ε > 0). This can be regarded as a variant of the Furutsu-Novikov theorem for the case of shot noise. As we show, the relation is still valid in the presence of additional independent noise. Furthermore, we derive an extension to Cox-process input, i.e. to colored shot noise. We discuss applications to particle detection and to neuroscience. Using the new relation, we obtain a fluctuation-response-relation for a leaky integrate-and-fire neuron. We also show how the new relation can be used in a remote control problem in a recurrent neural network. The relations are numerically tested for both stationary and non-stationary dynamics. Lastly, extensions to marked Poisson processes and to higher-order statistics are presented.