  In the recent paper arXiv:1606.02921, the two invariant actions for 6D
$N=(1,0)$ conformal supergravity were constructed in superspace, corresponding
to the supersymmetrization of $C^3$ and $C\Box C$. In this paper, we provide
the translation from superspace to the component formulation of superconformal
tensor calculus, and we give the full component actions of these two
invariants. As a second application, we build the component form for the
supersymmetric $F\Box F$ action coupled to conformal supergravity. Exploiting
the fact that the $N=(2,0)$ Weyl multiplet has a consistent truncation to
$N=(1,0)$, we then verify that there is indeed only a single $N=(2,0)$
conformal supergravity invariant and reconstruct most of its bosonic terms by
uplifting a certain linear combination of $N=(1,0)$ invariants.
