  We extend both the Hawking-Penrose Theorem and its generalisation due to
Galloway and Senovilla to Lorentzian metrics of regularity $C^1$. For metrics
of such low regularity, two main obstacles have to be addressed. On the one
hand, the Ricci tensor now is distributional, and on the other hand, unique
solvability of the geodesic equation is lost. To deal with the first issue in a
consistent way, we develop a theory of tensor distributions of finite order,
which also provides a framework for the recent proofs of the theorems of
Hawking and of Penrose for $C^1$-metrics [7]. For the second issue, we study
geodesic branching and add a further alternative to causal geodesic
incompleteness to the theorem, namely a condition of maximal causal
non-branching. The genericity condition is re-cast in a distributional form
that applies to the current reduced regularity while still being fully
compatible with the smooth and $C^{1,1}$-settings. In addition, we develop
refinements of the comparison techniques used in the proof of the
$C^{1,1}$-version of the theorem [8]. The necessary results from low regularity
causality theory are collected in an appendix.
