Is the money the mother receives from the state permissible, despite her official divorce from the father on paper, while their marital life continues for the purpose of her continued receipt of the grant allocated to martyrs' widows? And does this money become permissible after the father's death, or after correcting the divorce document?

The recipient of the grant is bound by the condition of the granting entity, and it is not permissible to use deception to obtain it. If the entity granting the pension stipulates that the widow should not marry, then she is not allowed to continue receiving the grant if she marries. However, if the condition of not marrying is merely a rumor and the pension of those who married was not discontinued, then there is no harm in the mother continuing to receive the grant. As for the issue of divorce, if the father pronounced it explicitly, then it is a valid divorce. But if he wrote it without pronouncing it, it does not take effect except by intention. If the father took back his wife during her iddah (waiting period), then that is valid, and according to some scholars, the revocation of divorce becomes valid merely by intercourse.