Is zakat due on a sum of money designated for Hajj, which has not been paid zakat for, and a full year has passed over it, knowing that part of this period it was a debt owed by others?

It is not permissible to delay Zakat (definition) beyond its due time except for a valid excuse. If one delays it, they commit a sin. The Hajj performed with money upon which Zakat was due before its extraction is valid. However, does the pilgrim incur an additional sin beyond the sin of delaying Zakat? This is a matter of dispute among scholars, based on the issue of whether Zakat is attached to the specific asset itself or to the owner's liability.

Whoever says that it is attached to the specific asset believes that the one who disposes of the money commits a sin. And whoever says that it is attached to the owner's liability believes that he does not commit a sin; this is the view of the Shafi'is. There is another opinion, such as that of Ibn Qudamah, who holds that it is permissible to dispose of the Zakat-eligible money, while still incurring the sin of delaying Zakat.

Therefore, the questioner's Hajj is valid, and he must pay the Zakat on the money that has completed its hawl (one lunar year), whether it is in his possession or a debt owed to him by others.