What is the ruling on the persistence of the smell of cleaning powder on clothes, and does that necessitate washing after going out in them?

A woman is not permitted to go out perfumed if she passes by non-mahram men, based on the Prophet, peace and blessings be upon him, who said: "If a woman perfumes herself and passes by people so that they may smell her fragrance, then she is such and such — he spoke a severe saying — meaning: an adulteress." As for laundry detergents with a noticeable pleasant scent that reaches non-mahram men, it is not permissible to go out with clothes cleaned with them. However, if the scent is faint and does not reach men, there is no harm, but it is preferable to avoid it altogether.

A woman is not required to perform ghusl unless she has perfumed her body. If she has perfumed her clothes, then she should change those clothes or remove the perfume from them.