Does a man's vow to pay a monthly sum from his salary for righteousness and good deeds include spending on good deeds that he, his wife, or his children perform, such as Hajj and Umrah, given his oversight in not considering that at the time of the vow?

As long as the one making the vow made an absolute dedication of the expenditure to acts of piety and did not restrict it, then the expenditure for Hajj and Umrah (definition) falls within the scope of the vow. It does not matter that this specific form of expenditure was not explicitly in his mind at the time of the vow, because it is part of the general category of expenditure for piety and good deeds.