This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.