  We apply the techniques developed in our previous work with Leung to study
smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded
perfect complex of locally free sheaves over a degenerate Calabi-Yau variety
$X$. In particular, if $X$ is a projective Calabi-Yau variety admitting the
structure of a toroidal crossing space and with the higher tangent sheaf
$\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free
sheaf over $X$, then we prove, using the recent results of Felten-Filip-Ruddat,
that the pair $(X,\mathfrak{F})$ is formally smoothable when
$\text{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.
