If always when cleaning the house from chametz before Pesach one already complies with the mitzva of ‘Tashbisu’ since this is a Biblical mitzva, in order to comply one has to have Kavana or intention to comply with the Mitzva. Is this correct? I don’t see people actually caring to recite a ‘Leshem Yichud’ before cleaning.

On question 3615 we addressed if one who eats chametz or drinks chametz during the thirty days before Pesach comply with the mitzva of Biur Chametz or destroying the chametz? If yes, should one have such Kavana (intention) in his mind to comply with a Biblical mitzva? It could make a difference in the keeping of hundreds or maybe even thousands of mitzvot. To what we answered; “Poskim disagree. Rema (O.H. 485: 1) rules that if one burns the chametz on the night after the Bedika, one complies with the mitzva of destroying the chametz. Chok Yaakov (ibid.), Mishna Berura (5) quote Maharik that since on the fourteen of Nissan, no chometz will be extant, even if the destruction of it was done during the prior thirty days, one has kept the mitzva when that day arrives. Horav Shmuel Kamenetzky Shlit’a (Kovetz Halochos – Pesach 1: 4: n: 5) indeed writes that following the opinion of the Minchas Chinuch that the mitzva of Tashbisu or destroying the chametz is a positive command, one indeed complies with the mitzva whenever one eats chametz during those thirty days. Horav Shlomo Miller’s opinion is that it depends on the different Halacha opinions.”