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Summary: We propose a new embedded finite element method to simulate partial differential equations over domains with internal interfaces. Our approach belongs to the family of surrogate/approximate interface methods and relies on the idea of shifting the location and value of jump interface conditions. This choice has the goal of preserving optimal convergence rates while avoiding small cut cells and related problematic issues, typical of traditional embedded methods. The proposed approach is accurate, robust, efficient, and simple to implement. We apply this concept to internal interface computations in the context of the mixed Poisson problem, also known as the Darcy flow problem. An extensive set of numerical tests is provided to demonstrate the performance of the proposed approach. [ABSTRACT FROM AUTHOR]