There are minhagim concerning the last paragraph of the song ’shalom aleichem’ as to whether or not ’tzeitchem’ should be sung. Could you please tell me the reason we sing it.

The Tosefet Shabbat tells a story of a Rabbi who erased the verse Tzeitchem Leshalom from his song book. He adds that this is the right thing to do. The Sha'arei Teshuva Orach Chayim 262 writes that one should say Tzeitchem Leshalom as the meaning is not to throw the Malachim out but rather that when they leave may they leave in peace. Rabbi Ro'i Margalit