Is the questioner—one of the deceased's needy sons—entitled to request a waiver of part of the money he borrowed from the one-third of his father's estate designated for needy children, knowing that he is paying off other debts?

If the father's intention was to dedicate one-third of his wealth for the benefit of his needy children, such that they possess the money and do not return it, then this is a void bequest because it is a bequest to an heir. This is based on the Prophet's (peace and blessings be upon him) saying: "Indeed, Allah has given every rightful person his right, so there is no bequest for an heir." Therefore, what the father bequeathed is to be distributed like the rest of his wealth, unless all adult, sane heirs agree to leave the money for this purpose.