The actual question is slightly more general than that in the title:



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> Let $p: U\to Y$ be a surjective étale morphism and $Y\to X$ be a finite morphism of schemes. Is there an étale cover $V\to X$ (surjective) such that the base change $p': V\times\_X U\to V\times\_X Y$ admits a section?
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If necessary, one can also assume that $Y\to X$ is surjective.


I think this is implicitly used in *Lecture Notes on Motivic Cohomology* by Carlo Mazza, Vladimir Voevodsky, Charles Weibel, Lemma 6.16. I don't know how to go through. I can prove this when $p$ is also a finite morphism and $Y=X$ and $Y\to X$ is the identity but not in general.


It would be very helpful to me to have a reference or an answer to this question.



Surjective étale morphisms étale locally split
We can work locally on $X$ and even (by standard limit arguments) assume that $X=\mathrm{Spec}(R)$ where $R$ is local and strictly henselian. Then $Y=\coprod\_{i=1}^{n}Y\_i$ where each $Y\_i$ is local and finite over $X$, in particular strictly henselian too. So $U\times\_Y Y\_i\to Y\_i$ has a section since it is étale and surjective.


