Nodal higher-order topological superconductivity from a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="script">C</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math> -symmetric Dirac semimetal

We analyze the topological properties of possible superconducting states emerging from a ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$-like, ${\mathcal{C}}_{4}$-symmetric Dirac semimetal, with two fourfold-degenerate points separated in ${k}_{z}$ direction. Unlike simplest Weyl semimetal for which all pairing orders are topologically obstructed and nodal, we show that obstruction semimetals crucially only exists certain symmetries. In particular, focus on odd-parity ${B}_{1u}$ ${B}_{2u}$ states, both can be induced by Ising ferromagnetic fluctuations. The inherit normal state, dictates these necessarily host four Bogoliubov--de Gennes (BdG) point nodes protected ${\mathbb{Z}}_{2}$ monopole charge. By Wannier state analysis, is higher-order nature. As result, rod geometry gapped surfaces, arcs Majorana zero modes exist regions hinges between BdG points. Fermi semimetals, stable against self-annihilation due to an additional $\mathbb{Z}$-valued charge ${\mathcal{C}}_{4}$ symmetry. find same also carried ${B}_{1g}$ ${B}_{2g}$ channels, where spectrum hosts bulk ``nodal cages,'' i.e., cages formed nodal lines, symmetry-preserving perturbations.