Recessed lights and ceiling fan
No... &amp; Yes... ?

perhaps you need to think this through a bit because ultimately you know (or will know) the wiring better than us.

In USA... Black "should be" HOT. White "should be" NEUTRAL. There should be a ground, you should use it. With that out of the way (noting that these are not always straight forward facts)...

Where does your supply come in? How do you intend to switch the lights on and off? If the supply goes through a switch, and your fan is currently wired to it, what do you suspect is going to happen when you switch that switch for the lights? (the fan will go off).

If the FAN has a remote, then it doesn't necessarily need a wall switch (but personally, I'd keep one anyways), however the lights will need a switch too... Right? You would need to add a switch for the lights. You can re-purpose the existing switch, but you'd still have to run a always HOT/NEUTRAL/GROUND to the fan (from a supply).

You have not mentioned how you intend to control the lights. If the supply side goes to the fan, then to the switch and back to the fan, then you can add lights off of the fan box provided you do not overload the box, or the circuit.  If the supply goes to the switch first, as is often the case, then you need to run another feed out of the switch box, to the fan, or the lights.