Which should be prioritized in spending: needy relatives—such as the wife’s needy mother and her children who wish to study—or less needy relatives—such as university student siblings whose father provides for them, and my mother who is well-off? And is it obligatory to spend on the wife’s mother even if my mother is angry about it?

Legally, you are not obliged to spend on your wife's mother and her siblings, even if she is your foster aunt. As for your mother and your siblings, you are not obliged to spend on them if they find sufficient sustenance. If they do not find it, you are obliged to complete it for them. For the obligation of maintenance, three conditions are required: the poverty of the person being spent upon, the ability of the spender, and the existence of a kinship that permits inheritance. If you wish to donate to your aunt and her children secretly, there is no harm, provided that your mother and siblings are first self-sufficient.