In this article, we consider additive functionals $ζ_t = \int_0^t f(X_s)\mathrm{d} s$ of a càdlàg Markov process $(X_t)_{t\geq 0}$ on $\mathbb{R}$. Under some general conditions on the process $(X_t)_{t\geq 0}$ and on the function $f$, we show that the persistence probabilities verify $\mathbb{P}(ζ_s < z \text{ for all } s\leq t ) \sim \mathcal{V}(z) ς(t) t^{-θ}$ as $t\to\infty$, for some (explicit) $\mathcal{V}(\cdot)$, some slowly varying function $ς(\cdot)$ and some $θ\in (0,1)$. This extends results in the literature, which mostly focused on the case of a self-similar process $(X_t)_{t\geq 0}$ (such as Brownian motion or skew-Bessel process) with a homogeneous functional $f$ (namely a pure power, possibly asymmetric). In a nutshell, we are able to deal with processes which are only asymptotically self-similar and functionals which are only asymptotically homogeneous. Our results rely on an excursion decomposition of $(X_t)_{t\geq 0}$, together with a Wiener--Hopf decomposition of an auxiliary (bivariate) Lévy process, with a probabilistic point of view. This provides an interpretation for the asymptotic behavior of the persistence probabilities, and in particular for the exponent $θ$, which we write as $θ= ρβ$, with $β$ the scaling exponent of the local time of $(X_{t})_{t\geq 0}$ at level $0$ and $ρ$ the (asymptotic) positivity parameter of the auxiliary Lévy process.