  We propose that doped Weyl semimetals with four Weyl points are natural
candidates to realize higher-order topological superconductors, which exhibit a
fully gapped bulk while the surface hosts robust gapless chiral hinge states.
We show that in such a doped Weyl semimetal, a featureless finite-range
attractive interaction favors a $p+ip$ pairing symmetry. By analyzing its
topological properties, we identify such a chiral pairing state as a
higher-order topological superconductor, which depending on the existence of a
four-fold roto-inversion symmetry $\mathsf{R}_{4z}$, is either intrinsic
(meaning that the corresponding hinge states can only be removed by closing the
bulk gap, rather than modifying the surface states) or extrinsic. We achieve
this understanding via various methods recently developed for higher-order
topology, including Wannier representability, Wannier spectrum, and defect
classification approaches. For the $\mathsf{R}_{4z}$ symmetric case, we provide
a complete classification of the higher-order topological superconductors. We
show that such second-order topological superconductors exhibit chiral hinge
modes that are robust in the absence of interaction effects but can be
eliminated at the cost of introducing surface topological order.
