For a density matrix $\rho_{AB}$ and some operators $A, B$, is there a way to express

$$\text{Tr}_A((A\otimes B)\rho_{AB})$$

using the reduced states $\rho_A$ and $\rho_B$ and operators $A$ and $B$? Note that when $A$ is the identity operator, this is indeed possible and one has

$$\text{Tr}_A((I_A\otimes B)\rho_{AB})= B\rho_{B}$$

as shown here.