Add a light switch to an existing switched outlet?
It is not possible to power a recessed light from the existing switch box. There is no neutral in this box, but only a line hot (always hot from the box for the existing receptacle) and a switched hot going to the existing receptacle.

The white wire in the switch box is not a neutral, but instead is being used to carry either the line hot or the switched hot from the switch to the receptacle.

Our tract house is wired this way and we have lamps plugged into the switched receptacle in each bedroom. I have considered getting a plug-in wall mounted lamp in one room (to replace a floor lamp) but haven't done it yet.