The speaker describes a moment when green comes back with the season and, almost against better judgment, belief returns with it. The word "again" matters: this has happened before, and the speaker knows it. What they believe in is called an "old untruth," something they've fallen for many times and probably will again.
The "melancholy lark" is the heart of it. Larks are usually pictured as joyful singers, and the speaker seems to be pointing at the habit of hearing sadness in a bird that may not be sad at all. Or the untruth might be the reverse, the cheerful spring story that hides some sorrow. The poem doesn't settle this, and judging by the wording, the speaker doesn't fully want to.
The mood is quiet and wry, a little tender. There's mild embarrassment at being so easily won over, but also a willingness to be won over anyway. The title, "Sucker," sums up that mix: the speaker admits to being fooled, and seems almost happy about it.