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Since every hyperelliptic curve $C$ of genus $g$ can be written as $y^2=f(x)$, is there an way to write an equation of a new curve $C'$ which has an etale cover over $C$ just using $x$ and $y$?

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Since nobody has answered this yet, here is my attempt, however Im not sure if it is exactly what you want. Hopefully it will at least lead you to a solution of your problem if you have not seen this kind of stuff already.

If your curve $C$ is defined over a field $k$ in which $f(x)$ splits as $f(x)=f_1(x)f_2(x)$, then we can define $C'$ via: $$C': y_1^2 = f_1(x) , y_2^2 = f(x).$$

Then the natural map $(x,y_1,y_2) \mapsto (x,y_1y_2)$ extends to an étale map on the compactifications.

However your question asks for $C'$ to "be written in terms of $x$ and $y$". If by this you mean present an affine patch of $C'$ which is a subset of $\mathbb{A}^2$, then $C'$ is at least birational to such an expression by the primitive element theorem and this shouldn't be too hard to write down - however it might be singular.

I hope this helps.

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