Clue: Lively little piece tutee played with behind library? (7)
Answer: BLUETTE
Explanation: Definition: Lively little piece
B[ritish] L[ibrary] + TUTEE. Bluette is a French word meaning a spark or flash, and by extension a short playful piece of music. It doesn't seem to be related to the colour blue, or the Blues as in music, (though there's a jazz piece by Dave Brubeck that uses the name); it's actually from a root meaning "white", cognate to beluga (as in the white whale) and the first part of the name Belarus (= "White Russia")