Yes, this is the modern statement of Weil's theorem of decomposition. It is a basic component of the theory of heights. For a more recent exposition see 2.7.15 in Bombieri and Gubler's *Heights in Diophantine Geometry*. 

If you look for a specific application of the theorem and its point of view, you should be aware of Bombieri's paper [*On Weil's "Theoreme de decomposition,"* Amer. J. Math., 1983]. There, Bombieri employs the theory of heights (Weil's theorem of decomposition and a theorem of Neron, cf. 9.3.10 in Bombieri-Gubler), to extend work of Sprindzhuk on the Hilbert irreducibility theorem and deduce a generalization of an old theorem of Runge stating the finiteness of solutions $(x,y) \in \mathbb{Z} \times \mathbb{Q}$ to $G(x,y) = 0$ for an irreducible $G \in \mathbb{Z}[x,y]$ whose leading homogeneous part is not proportional to a power of an irreducible polynomial.

Roughly, Bombieri's result is that if $f : C \to \mathbb{P}^1$ is a morphism from a curve over a number field $K$, and $P \in C(\bar{K}) \setminus f^{-1}(\infty)$ is a point with large height, then as the place $v$ of $K$ varies, each pole of $f$ is $v$-adically approached by $P$ with frequency proportional to the order of that pole. (Or, taking multiplicities into account, all poles of $f$ are approached with the same frequency.) To be a bit more precise, observe that $f(P)$ has a high local height at $v$ if and only if $P$ is $v$-adically close to *some* pole of $f$; Bombieri's theorem concerns the  portion of the height of $f(P)$ coming from a prescribed pole $Q$. Writing $P \sim_v Q$ to mean "$P$ is close to $Q$ at the place $v$" (in a sense that can be easily made precise), and assuming for simplicity that $P \in C(K)$, it states approximately that
$$
\sum_{v : \, P \sim_v Q} \log^+{|f(P)|_v} = \frac{\mathrm{ord}_Q (1/f)}{\deg{f}} \sum_v \log^+{|f(P)|_v} + O \Big( \sqrt{\sum_v \log^+{|f(P)|_v}} \Big)
$$
for every pole $Q$ of $f$, with the implied constant depending only on $f$.

This is itself very much in the spirit of Weil's theorem of decomposition. 

A modern exposition of this result of Bombieri's  and its consequent generalization of Runge's theorem is presented in the chapter 9 on Neron-Tate heights in Bombieri and Gubler's book. There, it is used to give an essentially algebro-geometric proof of the Hilbert irreducibility theorem.