Let $X$ be a smooth complex projective variety and $p:Y\to X$ be a locally trivial in analytic topology $\mathbb CP^k$-bundle. Suppose we have a line bundle $L$ on $Y$, restricting to $\mathcal O(1)$ on $\mathbb CP^k$-fibres.

Question. Is it true that there is a line bundle $L'$ on $X$ such that $p^*L'\otimes L$ is very ample on $Y$?

I think one could be able to prove this using Serre's (+ Kodaira?) vanishing, but I can't prove it so far.

Idea. Here is an idea of how one could try to solve this. So, first all, one can take $L''$ such that $p^*L''\otimes L$ is ample on $Y$. Next, one can try to use Kodaira vanishing, it says that $K_Y\otimes p^*L''\otimes L$ has zero higher cohomology. Now, we could take the line bundle $K_Y\otimes (p^*L''\otimes L)^{k+2}$, and this bundle will restrict to each fibre as $\mathcal O(1)$. By taking $L''$ positive enough, it should be possible to make $K_Y\otimes (p^*L''\otimes L)^{k+2}$ ample. I think that since this bundle is ample and its higher cohomology vanish, by Grothendiek-Riemann-Roch it will have a lot of sections (especially if $L''$ is very ample and has a lot of sections). This seems to be not far from proving that $K_Y\otimes (p^*L''\otimes L)^{k+2}$ is very ample... And I guess this bundle is $p^*L'\otimes L$ for some $L'$?