If I used Mewtwo's "Barrier" attack last turn and this turn I play Pokemon Ranger, does that clear the effect so I can use "Barrier" again this turn?

No, Barrier does not "place an effect" to prevent its use on the next turn (unlike attacks such as "Volcanic Heat" which says "this Pokemon" cannot attack next turn). Instead, the attack does a Game State check to see if it was used last turn. Since there is no effect to clear, Pokemon Ranger won't allow you to use Barrier two turns in a row.

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