We investigate a higher-order finite element method on a partitioned domain $\Omega$ with curved interfaces $\Gamma$ that is inspired by the hybridizable discontinuous Galerkin methods. The method uses Raviart-Thomas elements and piecewise polynomials for the dual and primal variables, respectively, and enhances mortar mixed methods by incorporating stabilization terms. This ensures the well-posedness of the patch problems for the Poisson problem and the reduced systems for the hybrid variable are symmetric and positive definite. In the difference to DG methods the presented analysis of the proposed finite element method is based on a corresponding variational formulation and a perturbed Galerkin method. We have identified a consistent variational formulation in which also the divergence has distributions on $\Gamma$ and the traces of the FE functions on $\Gamma$ are replaced by $L^2(\Gamma)$ distributions. By showing inf-sup conditions we can bound the discretisation error by the sum of the best approximation error and a consistency error contribution. The analysis is explicit in the "stabilisation parameter" $\tau$ of the formulation, which leads to a reduction of the convergence rates by 1/2 for large $\tau$ values. Numerical experiments with a finite element discretisation of high-order on curved quadrileral meshes confirm the error analysis for small values of $\tau$ and, to some extend, for large values, where unlike in the analysis the errors are bounded even for large values of $\tau$. We expect that the new analysis, which is based on a variational formulation with $L^2(\Gamma)$ distributions, to transfer to hybridizable discontinuous Galerkin methods. Furthermore, we expect that enhancing of mortar mixed methods with the stabilisation terms will lead to weaker conditions on meshes and polynomial degrees for mortar coupling.