There are several applications in computational fluid mechanics that require solving many times the flow equations, and one of them is the optimisation of aerodynamic profiles. A fast numerical solution of the flow problem is thus peremptory, and a means to achieve this is using reduced order models (ROMs). In particular, we propose to use ROMs based on proper orthogonal decomposition (POD) and having a finite element (FE) approximation as full order model (FOM). As usual for nonlinear problems, hyper-reduction (HROM) is also needed to have an efficient model. ROMs that use the FOM equations are often called intrusive. In contrast, there are also ROMs purely designed based on existing data of high fidelity and using machine learning (ML) techniques; these are called non-intrusive models. In this work we propose to design a hybrid model, essentially of intrusive type but incorporating a correction term based on ML techniques. The idea is to start from a purely POD-based ROM, projecting the equations onto the ROM space, and then add a nonlinear correction that depends on the ROM unknowns to enhance the final ROM model. This correction is based on the fact that we do have some available high fidelity data, namely, the snapshots. Thus, the correcting term is built as an artificial neural network (ANN) constructed with the snapshots as training set, i.e., considering that the loss function is the norm of the difference between the snapshots projected onto the ROM space and the outputs predicted by the model [1]. Other information from the FOM model could also be incorporated. The resulting hybrid ROM has a significant higher accuracy than the original intrusive one, correcting both the ROM and the HROM approximations. When applied in the context of shape optimisation, the correction turns out to be crucial not only to obtain better optimal points, but even to be able to approximate the correct (FOM-based) ones.