Is leaving overtime for two months to complete studies considered cutting off one's livelihood and a violation of the rights of the wife and son, knowing that this may lead to a better future?

There is no harm for a Muslim in elevating his scientific level and improving his material and moral conditions. Your return during the vacation to take an exam and raise your scientific level does not mean cutting off your provision, and the estimation of provisions is in the hand of Allah.

If your wife is outside the country where you reside, it is not permissible for you to be absent from her for more than six months except with her permission. If she is in the same country where you wish to complete your studies or exam, then you must go to her during the vacation period, combining both interests (maintaining ties with your family and completing your studies).

It is advisable to balance these matters and to kindly approach the official in charge of the additional work to obtain permission for absence.