Are there any results describing the $THH$ of spectra of the form $F(X, E)$ where $X$ is a space (say, finite CW) and $E$ is a (nice enough) ring spectrum?

I'm happy to put various (further, or other) restrictions on $X$ and $E$. I'm also happy to consider specific examples for $E$ (e.g. $H\mathbb{Z}$, $KU$, $MU$, $\mathbb{S}$ etc).

I'm hoping for something similar-ish to the result that $HH_*S^*X$, the Hochschild homology of the singular cochains on $X$, is equivalent to $H^*LX$, the cohomology of loop space. That already has a $THH$ analog in the Blumberg-Cohen-Schlichtkrull theorem, which describes the $THH$ of certain Thom spectra in terms of Thom spectra over loop space.

I would also be interested in "cyclic" results, i.e. some kind of topological analogs of the fact that the cyclic homology of $S^*X$ is the $S^1$-equivariant cohomology of $LX$.