Interactions between a relativistic-intensity laser pulse and a cone-wire target are studied by changing the focusing point of the pulse. The pulse, when focused on the sidewall of the cone, produced superthermal electrons with an energy >10 MeV, whereas less energetic electrons approximately 1 MeV were produced by the pulse when focused on the cone tip. Efficient heating of the wire was indicated by significant neutron signals observed when the pulse was focused on the tip. Particle-in-cell simulation results show reduced heating of the wire due to energetic electrons produced by specularly reflected light at the sidewall.