How should a man distribute expenses between his two wives when traveling, given that one has children and the other does not, and is he required to inform each wife of the amount of the other's expenses?

It is not obligatory for a man to give his two wives or his children the same amount of maintenance. Rather, he must provide maintenance to each of them according to their needs and circumstances, in an amount that suffices them. He is not obliged to inform one of his wives how much he spends on the other, and not informing her might be better.