  Analysis of higher-order correlation functions has become a powerful tool for
investigating interacting many-body systems in quantum simulators, such as
quantum gas microscopes. Experimental measurements of mixed spin-charge
correlation functions in the 2D Hubbard have been used to study equilibrium
properties of magnetic polarons and to identify coherent and incoherent regimes
of their dynamics. In this paper we consider theoretically an extension of this
technique to systems which use a pinning potential to reduce the mobility of a
single dopant in the Mott insulating regime of the 2D Hubbard model. We find
that localization of the dopant has a dramatic effect on its magnetic dressing.
The connected third-order spin correlations are weakened in the case of a
mobile hole but strengthened near an immobile hole. In the case of the
fifth-order correlation function, we find that its bare value has opposite
signs in cases of the mobile and of fully pinned dopant, whereas the connected
part is similar for both cases. We study suppression of higher-order
correlators by thermal fluctuations and demonstrate that they can be observed
up to temperatures comparable to the spin-exchange energy $J$. We discuss
implications of our results for understanding the interplay of spin and charge
in doped Mott insulators.
