Is it permissible for a divorced woman with children, whom a man married as a second wife without informing his first wife, to tell the first wife about her existence, in order to ensure that justice is upheld, knowing that the husband lives far away and she does not want him to be forced to lie?

A husband should be chosen based on his religiosity and good character, and this knowledge is gained by inquiring about him, not by emotion. If such a person is found, then it is good. As for the suitor residing far away, if you desire your full right to overnight stays and equitable division (qasm), then marriage is not advised unless his first wife is informed, to ensure justice and avoid problems, especially since concealing the marriage makes it difficult to fulfill rights. However, if you waive your right to equitable division and are content with his occasional visits, then marriage to him is permissible without informing his wife. It is not for you to inform his wife; rather, this is his concern, for you are entrusted with his secret. If you marry him and observe any shortcomings from him in the rights of the first wife, then you must advise him to be just.