A new local pseudopotential, called the "evanescent core" pseudopotential, has recently been proposed for sp-bonded metals. It is fitted to dominant density parameters of the solid state (valence, average equilibrium valence electron density, and interstitial valence electron density), and it yields an overall good description of physical properties such as binding energies and bulk moduli, in the framework of second-order perturbation theory. The potential, therefore, takes into account the atomic structure of the metal beyond the stabilized jellium model or structureless pseudopotential model. We present applications of the pseudopotential to surfaces and clusters of Na, Mg and Al, specifically: (i) Band-structure effects on surface tension and work functions; (ii) Cohesive energies and optimized structures of small clusters with two and six atoms (the latter with octahedral symmetry). The results are compared with those of the stabilized jellium model.