There are some surprising misunderstandings to clear up here.

Robert B. Waltz: The video Mr. Red links doesn’t claim that ‘Sumer’ is the earliest English song – it clearly isn’t – but that it’s the earliest surviving song in English with a complete text, which it is. (‘Mirie it is’ is earlier, but is incomplete.) … “several have said that the notation on "Sumer" is too modern to be used in a manuscript as old as the Harley MS. Some think the scribe wrote the arrangement; others think he copied it. If he composed it, he might write it down wrong”. Its date is on the cusp of the change from non-mensural to mensural music. It seems clear that this was written non-mensurally, then later amended to be mensural, which is when 3 notes were changed. It isn’t written down wrong – see my comments to Griska below.

Grishka wrote: “As for the theory mentioned in the OP, I am not convinced. The fourth note said to have been erased is a) not visible at all, and b) harmonically wrong, inconsistent with the rest of the song. Thus the entire idea collapses … I'm not convinced. A "d" on beat 4 without proper stepwise progress would be irregular.” This is wrong on all counts. A picture in the article originally posted here shows the manuscript as it stands on the left, digitally altered on the right to show the notes before they were altered by a later scribe (or perhaps the original scribe at a later date). We see clear and unambiguous erasure marks for notes 1, 3 and 4, note 2 being the same. As the article explains, the original notes are a return of the distinctive cuckoo call from earlier in the song, its reiteration changed by the amendment of notes. The originally written notes are c’’ a’ c’’ a’ (none of these notes are d’’ or d’ as you state). These notes are sung against, in the other voices respectively: c’’ a’ – e’ d’ – c’ d’ – g’ f’ – c’ d’. Harmonically, this is completely correct. Thus the original notes make sense both harmonically and artistically as a repeat of the cuckoo call. Not only this, with all voices singing, the restored cuckoo call cascades through the voices in a very effective way.