Let $\mathcal{C}$ be a small finitely bicomplete category.
If every object of $\mathcal{C}^{\text{op}}$ is a power of an object $x$, can we say that every object in $\mathcal{C}$ is a copower (is this what such a thing is called, taking the coproduct of an object with itself many times?) of $x$? I am a bit confused...And if that is the case, do we not have to change $x$ to something that actually lives in $\mathcal{C}$ rather than in $\mathcal{C}^{\text{op}}$?
Thanks
First, note that taking the opposite of a functor $F:A\to B$ sends the limit of $F$ to the colimit of $F^{\mathrm{op}}$. Your assumption is that every object $y$ of $\mathcal{C}^\mathrm{op}$ admits the structure of a limit of the constant functor $\Delta_x:\kappa\to\mathcal{C}^{\mathrm{op}}$ where $\kappa$ is a small discrete category. Now $\kappa^{\mathrm{op}}=\kappa$, so $\Delta_x^{\mathrm{op}}:\kappa\to \mathcal{C}$ has colimit $y$. In response to your confusion, it's convenient to see an opposite category as having the exact same objects as the original category. Then $x$ is in both places. If you find this convention artificial, then you will indeed have to replace $x$ with $x^{\mathrm{op}}$.
You can say copower, sure. You can also, and equivalently, say that every $y$ is a "tensor" of $x$ with some set; both of these are standard terminology from enriched category theory.