The poem is a tiny snapshot of the end of a long day for someone who cooks. She has a headache, and coffee is the remedy, which suggests she is tired and pushing on anyway. Then night "comes for the cook," a phrase that sounds almost like a threat, as if darkness and exhaustion are hunting her down.
The second part is the most striking. "Her art is no more / after supper." The poem calls her cooking an art, which gives dignity to work that often goes unnoticed. But the meal is eaten, and the art disappears. Nothing is left to hang on a wall or put on a shelf.
The speaker seems to watch from a distance, with quiet sympathy rather than loud emotion. The title, "Cooked," plays on the idea that she is both a cook and worn out, done in by the day. The mood is tired, gentle, and a little wistful.