Is it permissible for someone who paid monetary expiation for their oath to the mosque, then learned that this was not valid, to retrieve the money and pay it to the poor and needy?

Expiation is not valid if it is paid to those who are not indigent, such as for the construction or maintenance of a mosque. The one who paid the expiation has the right to reclaim it if it is still available. However, if the expiation was paid to the mosque's caretakers (ahl al-masjid) with the belief that they are entitled to it, or with the belief that they will disburse it to those who are entitled, then it is valid, and there is no need to reclaim it. Doubt after it has been paid does not affect its validity.