Is asking for manat allowed in our shariah? And if so, what is the proper way to do it?

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Yes it is allowed as per the sharia ruling

2650. While making a vow, a formula declaration has to be pronounced, though is not necessary that it should be in Arabic. If a person says: "When the patient recovers from his ailment, it will be obligatory upon me to pay $10 to a poor man, for the sake of Allah," his vow will be in order.