Recommended Resources / Literature Search Terms for a Solutions to a Specific Kind of Multi Harmonic Signal Structure
If I understand this problem correctly you have access to 2 signals:

Noise Signal - $ w \left[ n \right] $. It is composed of a linear combination of harmonic signals. Something like $ w \left[ n \right] = \sum_{i}^{m} {a}_{i} \sin \left[ 2 \pi \frac{ {f}_{i} }{ {f}_{s} } n + {\phi}_{i} \right] $.
Input Signal - $ y \left[ n \right] $ which is composed of the signal of interest $ x \left[ n \right] $ and the same harmonics as in $ w \left[ n \right] $ just with different weights. Something like $ y \left[ n \right] = x \left[ n \right] + \sum_{i}^{m} {b}_{i} \sin \left[ 2 \pi \frac{ {f}_{i} }{ {f}_{s} } n + {\psi}_{i} \right] $.

The objective is to estimate $ x \left[ n \right] $ given those signals.
Since we can model $ y \left[ n \right] = x \left[ n \right] + \left( h \ast w \right) \left[ n \right] $ we can utilize the adaptive filter framework and solve this in highly efficient manner.
Remark
In real world each observed signal would also have some kind of measurement noise we can model as AWGN. But let's leave it a side at the moment.