Is it a legitimate request for a wife to ask for equality between herself and her husband's children in expenditures, knowing that the husband spends most of his salary on his children and she feels embarrassed to ask him for her needs?

Equality in spending is not obligatory between wives, nor between a wife and children from another woman. The husband is obligated to spend on his wife in a reasonable manner that suffices her. She has no right to object to his spending on his children or to demand equality, nor does she have the right to demand a special allowance unless the husband voluntarily provides it.