We consider the Poisson problem in a domain with small holes, as a template for developing efficient and accurate numerical approximation schemes for partial differential equations defined on domains with low-dimensional inclusions, such as embedded fibers. We propose a reduced model based on the projection of Dirichlet boundary constraints on a finite dimensional approximation space, obtaining in this way a Poisson problem with defective interface conditions. We analyze the existence of the solution of the reduced problem and for arbitrarily small holes we prove its convergence towards the original problem, the rate of which depends on the size of the inclusion and on the number of modes of the finite dimensional space. The numerical discretization of the reduced problem is addressed by the finite element method, using a computational mesh that does not fit to the holes in the framework of a fictitious domain approach. We propose a stable, optimally convergent and robust formulation with respect to the hole size that exploits an augmented Galerkin formulation based on the addition of suitable non-polynomial functions to the finite element approximation space. The properties of the discretization method are supported by numerical experiments.