Put a layer of glue on one cut end of one of the long 1 x 4s. Butt the side of one of the short 1 x 4s against it (to form an L-shape) and drive two 6d nails through the outside of the short 1 x 4 into the end of the long 1 x 4. Use the damp cloth to wipe away any glue that oozes out.
Repeat with the other short 1 x 4 and another long 1 x 4.
Position the two assemblies to form a box. Put a layer of glue on the remaining cut ends of the long 1 x 4s, then butt the short 1 x 4s against them as before and nail through the outside of the short 1 x 4s into the ends of the long 1 x 4s. Wipe off any excess glue with the damp cloth.
Run a layer of glue around the top edge of all four boards, then position the thin plywood panel in place and secure it with the brads. This forms the bottom of the box. Wipe off any glue that oozes out. Let the assembled box dry.
Flip the box over so the open end faces up. Lay the large plywood panel on top, positioning it so the back edge is even with the back edge of the box and it overhangs the sides evenly. This forms the lid/writing area of the desk. Position the hinges along the back edge, marking the screw holes on the top of the lid and the back of the box. Drill the holes for the screws. Screw the hinges to the lid and box with the screwdriver. This completes your box with a hinged lid.
Put a layer of glue on the first 3 inches of the flat side of one of the remaining 1 x 4s, which will become the desk legs. Place the leg against one side, with one edge of the leg aligned with the vertical edge of the box and the cut edge aligned with the top edge just under the lid. Drive two 8d nails through the leg and into the edge of the box between the nails you drove into the box's joints in the previous steps. Wipe off any excess glue.
Repeat the process with the other three legs.
Sand the desk lightly. Apply the finish according to the manufacturer's instructions. Allow the finish to dry.