Você precisa GROUP BY todos os campos que deseja manter:
SELECT songs.id, songs.song, songs.artist,
AVG(score.score * 1.0) AS AvgScore
FROM songs
LEFT JOIN score
ON score.id=songs.id
GROUP BY songs.id, songs.song, songs.artist
ORDER BY songs.id, score DESC
Alternativamente, você pode simplesmente fazer isso:
SELECT songs.id, songs.song, songs.artist,
(SELECT AVG(Score) FROM score WHERE score.id = songs.id) AS AvgScore)
FROM songs