  We study a superconducting Kitaev ring pierced by a magnetic flux, with and
without disorder, in a quantum ring configuration, and in a rf-SQUID one, where
a weak link is present. In the rf-SQUID configuration, in the topological
phase, the supercurrent shows jumps at specific values of the flux
$\Phi^*=\frac{hc}{e}(1/4+n)$, with $n\in\mathbb{N}$. In the thermodynamic limit
$\Phi^*$ is constant inside the topological phase, independently of disorder,
and we analytically predict this fact using a perturbative approach in the
weak-link coupling. The weak link breaks the topological ground-state
degeneracy, and opens a spectral gap for $\Phi\neq \Phi^*$, that vanishes at
$\Phi^*$ with a cusp providing the current jump. Looking at the quasiparticle
excitations, we see that they are Anderson localized, so they cannot carry a
resistive contribution to the current, and the localization length shows a
peculiar behavior at a flat-band point for the quasiparticles. In the absence
of disorder, we analytically and numerically find that the chemical-potential
derivative of the supercurrent logarithmically diverges at the
topological-to-trivial transition, in agreement with the transition being of
the second order.
