The completed grid (with clashing squares containing first Across letters, then Down letters):
And from averaging the letters in each square (top to bottom, left to right), the answer is spelled out:
Bold clues are unsolved. Red bold shows the four words that are to be removed. Blue bold is words that look useless as I currently understand the clues, but don't seem to be part of the puzzle gimmick.
The theme, as I currently understand it,