Is it permissible to count the amount I lent to my poor relative who is struggling to repay it as part of the zakat money I possess?

Whoever is obligated to pay Zakat must do so immediately. It is not permissible to delay it except for a valid need. Paying it in installments is not sufficient. Therefore:

If the charities they gave you were from obligatory Zakat, then they erred by asking for it to be distributed over months.

If they were voluntary charities, then there is no objection to that.

You erred by disposing of the money without permission and by lending a portion of it to your relative. You are responsible for this money, and it is not permissible to consider it from your Zakat because forgiving a poor person's debt with the intention of Zakat is not valid.