From the switch new 2 wire cable is run to the new light location. Wiring for a new under cabinet light fixture.

How To Add A Light
Wiring a new light fixture and switch. Cut out the traced hole with an oscillating saw. These connect to the corresponding black and white electrical wires that run from. Now finish installing the light fixture to the ceiling box. The same is true for lights or switches in walls. Ceiling fixtures can vary from chandeliers for a formal dining room a new light and fan combination for a family room or a new look in a recently remodeled room. A clean work area will improve your ease of installation.
Two wire cable runs from the combo to the light fixture and the switch output is connected to the black wire running to the fixture hot terminal. When you change the fixture the national electrical code requires you to install an electrical box and update the wiring method to the current code. However the connections must be made within an approved electrical box. Wiring a new switch and light from an outlet in this diagram power is taken from an existing wall outlet for a new switch and light fixture. Twist the two ends together three or four times with a pair of pliers. If the light you are wiring is in the ceiling you should clean any cobwebs and dust before attempting to install your new fixture.
Look at the wires on the back of the new light fixture and note black insulation on one and white on the other. In the ceiling box you should have two wires that are not yet connected. The switch comes first in the new part of the circuit and new 2 wire cable is run to it from the outlet. Trace the mounting box on the ceiling where the light fixture will be mounted. The black wire from the fixture and the black wire from the switch. Determine a location for the light switch.
At the light it connects to the neutral terminal. The source neutral wire is spliced to the neutral on the receptacle half of the combo device and to the white cable wire running to the light. Before doing any electrical work always shut off power to the house at the main breaker. Connect those two wires together tighten a wire nut over them and tape them. Place the bare end of the black wire from the cable in the wall with the bare end of the black wire from the light fixture. It should be 48 from the floor.
The old electrical wiring is still acceptable as long as the insulation on them is intact. The two wires in the switch box is basically a loop of the power wire for the existing light fixture which brings the circuit power wire to the switch and then returns the power from the switch to the light fixture when the switch is turned on. Dress up a room or give it a whole new appearance by learning how to change a ceiling fixture on your own.