I'm going to assume the character in question is a Soulbow, because a Soulknife can't use Manyshot with a mind blade.
This effect deals an extra 1d8 points of damage to the next living, nonmindless target he successfully hits with a melee attack (or ranged attack, if he is using the throw mind blade ability).
It's also worth pointing out that the Soulbow's mind arrow class feature has the line
A mind arrow can be charged with a psychic strike as if it were a mind blade.
Technically, this doesn't remove the melee-only restriction on psychic strike. This, however, is a particularly obtuse reading of the rules, so I'm going forward assuming that a Soulbow can, in the general case, apply psychic strike to ranged mind arrow.
As you point out, both psychic strike and Psionic Shot only apply to your next attack after doing a thing (spending a move action, for psychic strike; expending your focus, for Psionic Shot). Since both abilities only apply to a single attack, the natural question is this: is Manyshot one attack, or multiple attacks?
The Player's Handbook defines an attack as
Any of numerous actions intended to harm, disable, or neutralize an opponent.
The outcome of an attack is determined by an attack roll.
This connects a single attack with a single roll, which would seem to support reading Manyshot as a single attack. Manyshot itself, however, says that
Damage reduction and other resistances apply separately against each arrow fired.
which is the behavior of multiple, separate attacks.
Either way, however, the result is the same. Either Manyshot is a single attack, in which case the bonus damage gets applied a single time, or it is multiple attacks, in which case the the bonus damage gets applied only to the first arrow, because psychic strike and Psionic Shot both only apply their damage to one attack (per move action or psionic focus spent). The only difference between the two abilities is that Psionic Shot gets spent even if you miss, whereas psychic strike only gets spent if you hit.
For either ability to apply to all the arrows, they'd need to be worded something like "Your attacks this round deal +2d6 damage," if you think Manyshot is multiple attacks, or "Your arrows deal +2d6 damage on your next attack," if you think Manyshot is a single attack. As it stands, however, I can't see a reading of these abilities that would let either Psionic Shot or psychic strike apply more than once to a Manyshot volley.
How Should You Rule It?
So the rules say no, but should you? That, I think, depends entirely on the power level of your campaign. A Soulbow can put out very respectable damage, and...not much else. They have the skills to be a scout, kind of, though no trapfinding. And once damage dealers start optimizing, things look bleaker for the Soulbow; there aren't a huge bevy of options for a WIS-based damage dealer, unlike some Barbarian with a greatsword. (People often say extra arms gives a Soulbow extra mind arrows; reading "must have a free hand" as "free hands give extra attacks" is a bit, uh, blatantly unfounded, but it does show that people often want Soulbow to be a bit better than it is.)
If the Soulbow is consistently outperforming other damage-focused characters, then you should probably enforce the rules as they're written (though if you do, give the Soulbow the option to swap out the relevant feats). If the Soulbow is struggling to contribute, on the other hand, then I say let Manyshot double/triple/quadruple damage from Psionic Shot and psychic strike. Multiple feats and setup in the form of a move action + psionic focus is a steep cost, so a steep reward isn't unreasonable.
So I would rule this depending on the texture and power level of the party and the player within that party. If you're like me and you maintain houserules across campaigns, however, I'd say Psionic Shot and psychic strike only apply once, and then maybe help a struggling Soulbow in some other way (removing the race restriction on Shiba Protector, for example, or maybe expanding the list of magical properties that can get applied to mind arrows) that wouldn't so broadly affect other archers down the road.