The poem is a quiet look at someone near the end of life, seen from outside. A father, alone "in the west," is meeting with the end every day. The title, "Terminal," suggests a terminal illness, though it also hints at a final stop, like the last station on a line. The west carries the same sense, since it is where the sun goes down. The speaker doesn't say who "her" is, but the phrasing suggests a daughter watching, or at least a voice close to her.
The second stanza turns to an image: a tree in a corner that "would sing of water." The tree seems to stand in for the father, and perhaps for anyone dried out and waiting. It can only long for what it lacks. The word "would" makes the song feel hypothetical, something wished for rather than heard. The mood is hushed, bare, and sad, with a little tenderness in the idea that even a parched thing might still sing.