Modelling Unwanted Signal in a LMS Adaptive Filter
The LMS and many of the variants of Adaptive Filters (In the Linear System context) work in the following settings (Intuitive):


You have access to 2 signals.
One signal is the result of the other one when a Linear System is applied.


This sounds really limiting, yet in practice it is powerful and flexible.

In the settings you mentioned the most known and similar problem is the Echo Cancellation model.

Pay attention that your model can't be formed as a Linear System.
Namely the connection between the clean signal and the corrupted signal can't be described using Linear System.