In the Time Card screen, you need to first select More > Show Entry Details to display the Entry Details dialog with the Memo field in it. When using the Time Card screen, enter the time in the relevant cell against a project-activity combination. Core displays the Entry Details dialog where you can enter a memo, if needed. Clicking outside the cell or dialog saves that entry and memo.
Time entries with a memo display a small blue arrow on the bottom-left. You can hover the cursor on the cell and click that arrow to open the saved memo on the Entry Details dialog. Check Core Help Center for details.