How to set the parameters of a Hidden Markov Model that'll be used to correct the mistakes by a previous classifier?
As far i know u can't saything about hidden class, hidden class value at time t is 'some intermediate values of weighted values of all hidden classes'. 

see the point-  2) hidden state sequence in blog-
https://machinelearningstories.blogspot.com/2017/02/hidden-markov-model-session-1.html

so ur statement-   

"Hence let in my HMM, the hidden state Zi denote the true class of the i-th sample Ii, and Xi be the predicted class by C- is incorrect.  

How can u compare hidden class value with actual, u could have compared emission value and actual. 

You can try ensembling of HMM and Cs. I wonder how exactly you trying to use parameters from classification problem to time series/sequence (HMM).