Step 1: Work out the relationship between the first pair.
"Weed" used as a verb means to remove unwanted plants from a plot so that only the desired ones remain, and the object of that action is a "garden." So the relationship is: a verb meaning to remove unwanted material acts on an object that is thereby improved or made ready.
Step 2: Apply the same relationship to the second pair.
"Edit" as a verb means to remove errors, unwanted material, or excess content from something so that it becomes fit for its final purpose. We need to find what an editor typically edits in this sense.
Step 3: Eliminate the options that do not fit.
(a) Publisher is a person or organisation that puts out the finished, edited work, not the thing being edited.
(b) Editor is the person doing the editing, not the object of the action.
(d) Puzzle and (e) Music are not standard objects of the verb "edit" in the way "garden" is the standard object of "weed."
Step 4: Confirm the best match.
(c) News fits naturally: an editor edits the news before it goes to print or broadcast, removing errors and unwanted content, exactly parallel to weeding a garden of unwanted growth before it is presentable.
Step 5: State the answer.
The correct answer is option (c), News.