This is a bit of a broad medieval-scholasticism-cum-engineering question. In QFT, all masses "run" with energy (I assume they are your "mass function") by virtue of renormalization. Within each energy regime, they present differently. Your QFT texts should specify when and which are focussed on.
Typically, at very high energies, (UV), they appear in your "bare" Lagrangian, to compute perturbatively their RG "running", and you compute their running effect on propagators, among others. They are the current masses, resulting out of the Higgs mechanism, typically of the order of a few MeVs for light quarks. They are called "current" because they serve in current algebra high energy applications and chiral perturbation theory, typically for light pseudoscalar meson masses. They feature in parton distributions.
Running this running backwards to the IR, to the extent possible, you start transitioning to this odd thin regime where it is believed they "morph" into some sort of "constituent" masses after chiral symmetry breaking, but before confinement. Masses of a few Hundreds of MeVs. You already appreciate they cannot be the constituents of pseudoscalar mesons like the pions (which are, in fact, much lighter than them--and constitute the pseudogoldstons associated to the chiral symmetry that made them so heavy in the first place). Beyond hadron spectroscopy, they inform elastic (forward scattering) and total cross sections. Lattice gauge theory tries to bridge the current-constituent gap, effectively.
It's all in the math: read up your texts, and, if they are any good, they will each specify the type of mass you use in each context. A huge wrench kit. You just can't handle everything with a mere adjustable wrench.