I'm looking for a hypothetical scientific discovery that would make both things possible. I'm just looking for the most plausible scenario.
Experiments to demonstrate telepathy tend to fall narrowly in favour of it, albeit close to the margin of error, so the scientific discovery would potentially be something that amplified a latent existing ability. However whether this would be easier than a simple messaging system that we already have available to us is another matter- practically a phone with a headset is only narrowly differentiated from telepathy right now.
Clairvoyance again might be effectively emulated with technology - the ability to connect to remote cameras all over the world is a practical form of remote viewing. If you wanted a scientific discovery that facilitated it as a purely mental ability, then how about the telepathy one we already mentioned? Being able to pick up the view of a remote scene telepathically would be a logical and consistent way of seeing it. If you wanted to perceive an area with nobody in it, maybe allowing the telepathy to connect to other mammals might offer a similar advantage, although limited to the sense-data the animal in question has available, so colour perception, field of view and so on may be limited. What would be a lot more challenging ( and require a lot more structure in terms of worldbuilding ) would be to be able to see completely empty locations or other points in time.
What is the discovery that facilitates this? A genetic connection between individuals ( explaining why telepathy seems to be most common in families, between twins and so on ) could make sense. "Genetic radio, we call it. It isn't radio and it's not entirely genetic, but close enough." To engineer a telepathic bond between two individuals might involve a specific type of gene therapy, but it turns up in nature as well. Of course that wouldn't allow your animal clairvoyance above, that would require someone to be able to open themselves up to the signals coming from everywhere, then narrow in on individuals in a location. One might imagine that regardless of technology, this would be a challenging feat requiring a lot of focus.
We just need to learn more about how the brain works, so that we can build electronics that interfaces directly with it.
Once we have proper wifi equipment in our skulls, many things become possible.
Voluntary telepathy is just online messaging. Involuntary mind reading is harder, that would be hacking the victims brain. If that happened, I would think somebody should have a serious chat with the designers of that wifi equipment.
As for clairvoyance, you have the whole internet in your skull! Streaming remote webcams is only the start of what you can do!
If by telepathy you mean the ability to read another's mind, the principle is simple: any brain works based on electrical activity, signals being exchanged between neurons.
We already know that any electric signal can be detected, so to make telepathy real we simply need to be able to collect the signals emitted by a brain and read them with our brain.
This would require a translation table, telling which interaction in the scanned brain correspond to which thought in the scanner brain.