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 $LX$.

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    $\begingroup$ If $X$ is finite CW, then it is dualizable in spectra, so that $F(X, E)\cong E\tensor DX$, and then $THH(F(X,E))\cong THH(E)\tensor B^{cyc}(DX)$, where $D(X)\tensor D(X)\to D(X)$ is the dual of the diagonal. I don't know what is known about the $B^{cyc}(DX)$, but it seems approachable at least. $\endgroup$ Mar 30, 2020 at 2:19
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    $\begingroup$ This might also give a connection to the Blumberg-Cohen-Schlichtkrull result, since Atiyah duality gives that $\Sigma_+^\infty M$ and $Th(NM)$ are dual. $\endgroup$ Mar 30, 2020 at 2:26


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