Despite the praise heaped on this puzzle, I feel the preamble is badly written, confusing and misleading.
You apply the down clue words from initials (8, 6) to the WORDS removed from the across clues. Not to the phrase created by initials
Note the words are quite specific, they are chosen not just for their first letter....
This gives three replacement P___s. (But not for the three you mention, there are five.)
The final adjustment is a bit of nonsense, but its the only word that fits. And will seem very, very familiar.