Does this Esper use the psionic die?
Sort of.
You get a reservoir of Psi Dice. Your dice starts as a d4 and you get 2 at level 1. You also get -one- power.
A power is basically a Cantrip. But it can be Augmented. There are basic augments, like "Do more damage" that apply to any power, and there are power-specific augments. Like increasing the weight limit of Telekinesis, or using Telekinesis to grapple an enemy at range. . And you can assign psi dice to each augment you want (or the same one several times for more damage, as an example)
You assign Psi Dice to augments, manifest the power. Then roll any assigned Psi Dice. On a 1 or a 2, you lose that Psi Die.
As you gain levels, you gain more psi dice, but also they increase from d4s to d6s to d8s to d10s. And you still lose them only on a roll of 1 or 2.
The actual powers with 1 psi die are a -little- weaker than a 1st level spell. But you have a 50/50 keep your psi die, so it's like you didn't spend your spell slot at low level. But you're much more likely to risk several dice at once to augment your power a lot when you gain levels, which can cost you if you roll several 1s.
You recover Psi Dice during a short rest by sacrificing Hit Dice. Otherwise you get them all back at the end of a long rest.
So let's say Aldarc is a level 8 Esper. He's 4d8 reservoir and Telekinesis he wants to use on a displacer beast. Thedisplacer beast is threatening the party Archer, Maxperson. He looks at his power and decides not to focus on damage, but to get Max to safety. So he puts 3 Psi Dice into "Lift". Beasty makes a Strength save or gets lifted up to 15ft into the air 'til the start of Aldarc's next turn, plus takes TK's base damage of 1d8 bludgeoning. Then for a Bonus Action uses a Mind-Strike to add a little more damage (1d6+int from a ballistic strike launching a rock at the displacer beast, if he hits)
Once the tenta-cat is flailing around in the air, 15ft away from Maxperson, Aldarc rolls those 3d8. Gets a 1 and two 6s. He loses one Psi Die from his reservoir, and Max gets away to shoot at the displacer beast.
On the Displacer Beast's turn it... flails. It doesn't have a fly speed or anything so it just hangs out in the air and cant reach anyone. It's turn is wasted.
On the following turn, Aldarc gets to choose if he wants to roll 3 dice to maintain the Lift. But decides he doesn't wanna put dice on Lift, so lets go of the cat, which falls 15ft, then winds up the 3 psi dice into TK, and throws a 4d8 TK blast at the kitty. Rolls to hit, rolls damage, rolls those 3d8 to see whether he loses another dice. 4, 3, 7. Keeps his dice.
Etc.
Mind Strike is also cool... one of four different Bonus Action attacks that improves to 2 dice at level 10 to help offset the fact that powers don't auto-level at the rate cantrips do.