Is it permissible for my husband to secretly clap his hands loudly in the air after reciting Ayatul-Kursi at night?

It is recommended to recite Ayat al-Kursi before sleeping. As for clapping after reciting it, that is a reprehensible innovation, and there is nothing in the Sharia of Allah the Almighty concerning it. Rather, it is an act of the Sufis who practice rituals for which Allah has sent down no authority.