We introduce a data-driven approach to enhance the accuracy of non-intrusive Reduced Order Models (ROMs). In particular, we focus on ROMs built using Proper Orthogonal Decomposition (POD) in an under-resolved and marginally-resolved regime, i.e. when the number of modes employed is not enough to fully capture the system dynamics. We propose a method to re-introduce the contribution of neglected modes through a quadratic correction term, given by the action of a quadratic operator on the POD coefficients. Differently from the state-of-the-art methodologies, where the operator is learned via least-squares optimisation, we propose to parametrise it by a Multi-Input Operator Network (MIONet). This way, we are able to build models with higher generalisation capabilities, where the operator is continuous in space – thus agnostic of the domain discretisation – and can be a function of the problem’s parameters. Experiments on two standard benchmarks in fluid dynamics show that these features allow for the construction of small and accurate models. These outperform least squares-based models, and effectively improve the performance of POD-based ROMs in the context of sparse and scarce training data. The effectiveness in the sparse and scarce data regimes is especially important, as it indicates that the present methodology could be successfully employed in industrial problems of practical interest, where other traditional methods might fail.