  Entanglement entropy in causal sets offers a fundamentally covariant
characterisation of quantum field degrees of freedom. A known result in this
context is that the degrees of freedom consist of a number of contributions
that have continuum-like analogues, in addition to a number of contributions
that do not. The latter exhibit features below the discreteness scale and are
excluded from the entanglement entropy using a "truncation scheme". This
truncation is necessary to recover the standard spatial area law of
entanglement entropy. In this paper we build on previous work on the
entanglement entropy of a massless scalar field on a causal set approximated by
a 1+1D causal diamond in Minkowski spacetime. We present new insights into the
truncated contributions, including evidence that they behave as fluctuations
and encode features specific to a particular causal set sprinkling. We extend
previous results in the massless theory to include R\'enyi entropies and
include new results for the massive theory. We also discuss the implications of
our work for the treatment of entanglement entropy in causal sets in more
general settings.
