How to utilize user feedback due to miss-classification when correct class label is unknown?
This can be accomplished by a modification to multi-class cross-entropy.

We are faced with two types of supervision. First type is "data $i$ belongs to class $k$" denoted by $y_{ik}=1$, and second type is "data $i$ does not belong to class $k$" denoted by $\bar{y}_{ik}=1$. For example, for 3 classes, $y_i=(1, 0, 0)$ denotes that point $i$ belongs to class $1$, and $\bar{y}_{i}=(0, 0, 1)$ denotes that point $i$ does not belong to class $3$. Let $y'_{ik} \in [0, 1]$ denote the model prediction. The original cross-entropy for $K$ classes is:  

$$H_y(y')=-\sum_{i}\sum_{k=1}^{K}y_{ik}log(y'_{ik})$$.

This objective assigns loss $-log(y'_{ik})$ to $y_{ik} = 1$ to encourage the model to output $y'_{ik} \rightarrow 1$ leading to $-log(y'_{ik})\rightarrow 0$.

On the other hand, for the second supervision $\bar{y}_{ik}=1$, we want to encourage the model to output $y'_{ik} \rightarrow 0$. Therefore, loss $-log(1- y'_{ik})$ can be used to have $-log(1- y'_{ik})\rightarrow 0$.

Accordingly, second supervision can be combined with first one as follows:

$$H_{(y,\bar{y})}(y')=-\sum_{i}\sum_{k=1}^{K}y_{ik}log(y'_{ik})+\bar{y}_{ik}log(1-y'_{ik})$$

Note that supervision "data $i$ does not belong to classes $1$ and $2$" is also supported. For example, $\bar{y}_{i}=(1, 1, 0,...)$ activates both $-log(1 - y'_{i1})$ and $-log(1 - y'_{i2})$ to encourage the model to output less probabilities for classes $1$ and $2$, i.e. $y'_{i1} \rightarrow 0$, and $y'_{i2} \rightarrow 0$.