  In this note we extend the dimension theory for the spaces
$\widetilde{\mathcal H}_g^k(\mu)$ of twisted $k$-differentials defined by
Farkas and Pandharipande in [FP15] to the case $k>1$. In particular, we show
that the intersection $\widetilde{\mathcal H}_g^k(\mu) \cap \mathcal{M}_{g,n}$
is a union of smooth components of the expected dimensions for all $k\geq 0$.
We also extend a conjectural formula from [FP15] for a weighted fundamental
class of $\widetilde{\mathcal H}_g^k(\mu)$ and provide evidence in low genus.
