My mother has $1000 for one and half years, and I have $500 for the same period. Do I have to pay Zakah? If yes, how much is it? Can I pay it directly to the poor without giving it to the government? Am I late paying Zakah since I have to pay it every year - am I now late by 6months? Is there any way to correct this mistake?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Zakah is due on each of the two amounts you mentioned as each of them has reached the Nisaab which is the equivalent of the value of 85 grams of gold.

Zakah should be given as soon as owning the property a lunar year occurs because payment should not be delayed except for a clear advantage for those to whom it is given.

However, delay does not cancel the obligation to pay it.  So, you should now give the Zakah of your money and repent to Allah for the delay.

In addition, know that it is better that you take care of giving the Zakah to the poor and the needy yourself.  It is also better that the Zakah be given to those who deserve it among your relatives and the closer to you the better because it combines preserving kinship ties and Sadaqa.

Finally, know that when you delay Zakah from its fixed time, you don’t have to do anything in expiation.  But you should repent to Allah and pay the Zakah you missed.

Allah knows best.