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.

Bombieri's result states that if $f : C \to \mathbb{P}^1$ is a morphism from a curve over a number field $K$, and if $P \in C(\bar{K}) \setminus f^{-1}(\infty)$, then for each pole $Q$ of $f$ it holds
$$
\sum_{v : \, P \sim_v Q} \log^+{|f(P)|_v} = \frac{\mathrm{ord}_Q (1/f)}{\deg{f}} \sum_v \log^+{|f(P)|_v} + O \Big( \sqrt{\sum_v \log^+{|f(P)|_v}} \Big),
$$
where $P \sim_v Q$ means that $P$ is in a $v$-adic neighborhood of $Q$ depending only on $f$, and $\log{|\alpha|_v^+ := \frac{1}{[F:K]}} \log^+{|N_{F/K}(\alpha)|}$ for $\alpha$ belonging to an extension $F/K$. Intuitively, noting that the local height $\log^+{|f(P)|_v}$ of $f(P)$ at $v$ is large if and only if $P$ is $v$-adically close to *some* pole of $f$, this means that as $v$ varies, every pole is approached with frequency proportional to the order of the pole. This exactly captures the spirit of the 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.