You can't delete those. Just hide it by clicking that first icon on the header.
How can I delete a Social Inactive Chat message?

Many thanks for your response. I understand that I can only hide this inactive chat.
Is it because the user is banned, that's why I am unable to delete the chat? If this was a normal chat where the user was not banned, will I have been given the option to delete the chat message?
I ask this, because I used to have a similar experience like this in 2014 Ubisoft game forums where a banned user sent me a spam message in my visitor profile page and I was unable to delete it as there was no delete button. Only for non-banned users it was possible. However, having sent a request to a forum moderator at that time, he was able to remove it manually from their end.
I'm guessing, you don't have the power to do this as a Mod yourself? Do you believe a Chess.com Admin can delete it from their end?
Around 2 years ago I had this user called Online_ChessCoach message me. He claimed to be a FIDE Master.

Anyway, I never ever actually responded to his message. Nevertheless, this account has been closed for abuse shortly after seeing it. Apparently, it appears that he has been banned due to spamming the same chat message to other users of Chess.com.
I now find that this message is impossible for me to delete. Even if I add his account to the blocked list the message remains. The chat cannot be deleted. If I press on the Inactive Chat, his username along with the message still appears on the Chat tab.
I'd like to have this Chat message deleted, but I don't know how. I have not inquired about this previously and it is the first time I have posted on these forums since being a member here on Chess.com last 7 years.
Is this something only Chess.com can assist with? Surely there should've been a delete button from the users end.
Many thanks.