You know the reason for termination. It was your dual employment in the past. Then why should you get a termination letter showing the reason for termination? It is good if the reason is not mentioned because a termination letter with reason would cause stigma to you and there is no use of it. Just ignore the letter. But ensure that you get a relieving letter which shows that you have been relieved.
Now there is a failure of procedure in your case. You were issued a show cause notice. You should have responded to the notice also. Then what happend? There should be some further communication, like informing you that your dual employment is a misconduct and therefore, you are going to be terminated. There should also be a letter informing you the decision of the appointing authority that your service would be terminated and if you need to give any reply please show cause why your service should not be terminated. Without these steps, if the HR is terminating you, the termination will not be valid. You can challenge the termination before the appropriate authority. BUT do you want to defend it? Even if you defend the authority will only say that there is a procedure lapse and therefore, complete the procedure but will not entertain in the decision of termination. The reason for termination should be concealment of facts and that could be proved. If you move legally, you will agains lose your career.
One more thing connected with dual employment is that for any employer what matters is whether the employee is emplayable or not. If he is fit to our requirement, nobody would unearth the past. In your case also the employer had taken more than six months just to see whether you are fit for the organisation and the real reason for termination is not dual employment but they do not need you and for the sake of saying some reasons they are saying that you have dual employment. If that is the reason, they should have terminated you right in June or July 2023.