The aim of this paper is to unify the ideas and to extend to a more general setting the work done in (Dell'Accio et al., 2023 [14]) for a polynomial enrichment of the standard three-node triangular element (triangular linear element) using line integrals and quadratic polynomials. More precisely, we introduce a new class of nonconforming finite elements by enriching the class of linear polynomial functions with additional functions which are not necessarily polynomials. We provide a simple condition on the enrichment functions, which is both necessary and sufficient, that guarantees the existence of a family of such enriched elements. Several sets of admissible enriched functions that satisfy the admissibility condition are also provided, together with the explicit expression of the related approximation error. Our main result shows that the approximation error can be decomposed into two parts: the first one is related to the linear triangular element while the second one depends on the enrichment functions. This representation of the approximation error allows us to derive sharp error bounds in both and norms, with explicit constants, for continuously differentiable functions with Lipschitz continuous gradients. These bounds are proportional to the second and the fourth power of the circumcircle radius of the triangle, respectively. We also provide explicit expressions of these bounds in terms of the circumcircle diameter and the sum of squares of the triangle edge lengths.