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 $$ \sum_{v : \, P \sim_v Q} \log^+{|f(P)|_v} = \frac{\mathrm{ord}_Q (1/f)}{\deg{f}} \sum_v \log^+{|f(P)|_v} + O \Big( \sqrt{\sum_v \log^+{|f(P)|_v}} \Big), $$ with the implied constant depending only on $f$. Here I assume for simplicity that $P \in C(K)$ and write $P \sim_v Q$ to mean that $P$ is $v$-adically close to $Q$ in the sense of belonging to a $v$-adic neighborhood, depending on $f$. (This extends to all $P \in C(\bar{K})$ with a standard change of notation. )
Here is the intuitive meaning of this. Note that the local height of $f(P)$ at $v$ is large if and only if $P$ is $v$-adically close to some pole of $f$. Then as $v$ varies, each pole is approached with frequency proportional to the order of the pole. Or, putting this in different words, the contribution from each pole $Q$ to the height of $f(P)$ can be approximately read from the factorization of the function $f \in K(C)$. 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.