No, this should not be in the TMF. It should be filed within your quality management system and/or CAPA tracking system. External CAPAs need to be filed also so that you show oversight. Again, should be in the QMS.
I’m also interested in this chain as I have heard both suggestions. File CAPAs if they help tell the story of the TMF or show Sponsor Oversight. I have also heard not to file them and place in QA files like QMS. What about site level CAPAs?