The key to solving this common piece of The Completed Moon's Map letter puzzles is determining which letters to add to a character's name on the map screen. The secret lies within the story page for the selected puzzle (and in particular, the final line of the page).
For example, the last line in the "Agar" story is "Davies retired her dais". The pattern to recognize is that something about the smaller word at the end of the statement is removed/negated in regards to the larger first word in the statement. If we take this literally, the secret is to remove the letters that make up the smaller word from the letters in the larger word. In this case, "DAVIES" minus the letters "D", "A", "I" and "S" results in "VE" left over. So, when working on the Agar puzzle on the Moon's Map, you would type "VE" after "AGAR".
Once the correct letters are added to a name, a word needs to be unscrambled from all of the letters. Using the example above, "AGARVE" would be unscrambled to "RAVAGE". Unscrambling the word grants access to the next part of the puzzle.