We introduce a flux recovery scheme for an enriched finite element method applied to an interface diffusion equation with absorption. The method is a variant of the finite element method introduced by Wang $et$ $al$. in [20]. The recovery is done at nodes first and then extended to the whole domain by interpolation. In the case of piecewise constant diffusion coefficient, we show that the nodes of the finite elements are superconvergence points for both the primary variable $p$ and its flux $u$. In particular, in the absence of the absorption term zero error is achieved at the nodes and interface point in the approximation of $u$ and $p$. In the general case, pressure error at the nodes and interface point is second order. Numerical results are provided to confirm the theory.