Does the wife deserve "mut'ah" (a compensatory gift given to a divorced woman) if she requests a separation because she refuses her husband's marriage to another, knowing that he has not had children with her for three years because it is God's will and not due to a defect in her or in him?

A husband is permitted to marry a second wife, and this is not a legitimate religious reason that allows the first wife to seek divorce. If the wife requests a divorce for this reason, the husband has the right to refuse to divorce her, or to divorce her. If he divorces her, it appears she is not entitled to mut'ah (compensatory gift), because mut'ah is forfeited if the separation originates from the wife, or if she is the one seeking the divorce. This is because the wisdom behind mut'ah is to console the wife who is distressed by the separation, and such distress does not befall her if she is the one desiring the divorce.