An empty mark does more than record a fact — it changes the line's geometry. A line with an X in it is really two shorter lines, and the puzzle of “which clues belong to which segment” is often trivial to answer: a run simply does not fit where there is not enough room. Whenever the answer is forced, you can solve each segment as an independent miniature line.
Take a width-9 line with clues 4 2 and an X in the fifth cell. The left segment is exactly four cells wide; the right is four. The run of 4 does not fit on the right together with the 2 (that would need seven cells), so the 4 must occupy the left segment — and since the segment is exactly four wide, it fills completely:
Even when a clue is not fully forced, counting fits narrows things. If the first two clues cannot both fit to the left of the X, at least one lives on the right — and ordering is preserved, because clues never swap places. Assign what you can, then run the overlap method inside each segment using only the clues assigned to it.
A related pattern shows up at the walls, sometimes nicknamed mercury: a filled cell close to a wall, whose run is longer than the gap between them. The run cannot fit between the cell and the wall, so it must spill toward the middle — like mercury pushed along a tube — and the cells at the far end of its reach get crossed off. With clue 4 and a filled third cell in width 8, the run can start at cells 1-3 only, so it never reaches cells 7-8, and every placement covers cell 4:
Splitting is the technique that makes 20x20 and 25x25 grids tractable: by mid-solve, no line is really its printed length any more. Re-measure segments every time you place an X, and hard lines keep collapsing into easy ones.