In the Nusach Ari used by Chabad, Yedid Nefesh appears as a continuation of Pasach Eliyahu to be said before Mincha. The standard print of the Tehilas Hashem siddur explicitly instructs to say it before Mincha although it is ambiguous at best in the Alter Rebbe's Siddur, from the Siddur Torah Or on it is put squarely before Mincha.
However, many Chabad Houses I have attended will sing Yedid Nefesh right before Lecha Dodi. Is this simply a way of adding a nice tune, or is there a reason behind the practice related to the Nusach of the prayers?