This article concerns an extension of the topological derivative concept for 2D potential problems involving penetrable inclusions, whereby a cost function $J$ is expanded in powers of the characteristic size $\varepsilon$ of a small inclusion. The $O(\varepsilon^{4})$ approximation of $J$ is established for a small inclusion of given location, shape and conductivity embedded in a 2-D region of arbitrary shape and conductivity, and then generalized to several such inclusions. Simpler and more explicit versions of this result are obtained for a centrally-symmetric inclusion and a circular inclusion. Numerical tests are performed on a sample configuration, for (i) the O(\varepsilon^{4})$ expansion of potential energy, and (ii) the identification of a hidden inclusion. For the latter problem, a simple approximate global search procedure based on minimizing the $O(\varepsilon^{4})$ approximation of $J$ over a dense search grid is proposed and demonstrated.