I hope that you can help me because in fact I am not able to put up with this injustice any more.
When a husband conceals his second marriage, in most cases that leads to some kind of injustice for his new wife, as he will be confused and uncertain and will be afraid that he may do something that would lead to his first wife finding out that he has taken a second wife. This may lead him into a series of mistakes.
As you agreed to that at first, then you have to put up with what happens to you in some ways and you have to try to set things straight in other ways.
If your husband had an excuse -- in your view -- before his marriage to you was discovered, then he has no excuse now. What he has to do is to teach you and his first wife fairly with regard to spending the night. However many nights he spends there, he has to spend the same number of nights with you, and you have the right to demand this right which Allaah has obliged him to give and that Allaah has given to you. If he persists in refusing, then you have the choice: either you can accept your life with him and put up with it until Allaah grants you relief -- which is what we recommend -- or you can choose to leave him.
As the matter has been discovered, we advise you to appoint as an intermediary someone who has knowledge and is of good character, who can intervene between you to resolve your problem with him and make him do that which Allaah has obliged him to do of treating you and his first wife fairly, and registering your daughter officially. This is something that is essential. How can he accept for his daughter to remain like that, with no recorded lineage and with her rights exposed to loss?
Now the matter is up to you. Advise him and remind him of Allaah, and if he does not respond, then appoint as intermediaries wise people from among your family, or from among your family and his, to advise him and make him adhere to that which Allaah has enjoined upon him of treating his wives fairly and registering his daughter in the official records.
Ask Allaah to help and guide you and him. We ask Allaah to bring you together on the basis of good and to make it easy for you to do that which pleases Him.
And Allaah knows best.