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