All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, all perfect praise be to Allaah, Who guided you to Islam and we ask Him to make you firm and steadfast on His Religion until the day you meet Him.
We ask Allaah to reward you for being steadfast on the truth and make your husband and daughter a comfort to your eyes. There is no doubt that it is not permissible for a Muslim to establish an affectionate relationship with a marriageable woman. Hence, if your husband established a relationship with a girl, then he is wrong. However, since he mentioned that he abandoned this relationship, you should trust him, and it is not permissible to accuse him of re-establishing this relationship without evidence. You should also believe him about the debts, and a husband is not obliged to inform his wife about his investment or any other monetary issue.
Therefore, we advise you not to open the door of the whispers of satan because this may spoil your life and cause divorce between you and your husband, and if this happens, it is a calamity. Rather, you may frankly talk to your husband and agree with him about what you want him to be, like being a pious and righteous man, truthful, and the like.
As regards the supplications which you asked about, there are no specific supplications but you may supplicate with the verse in the Quran in which Allaah Says (what means): {Our Lord, grant us from among our wives and offspring comfort to our eyes.}[Quran 25:74]. You may also supplicate with the Prophetic supplication: "O Allaah make him steadfast and firm on Your Religion and guide him." (a supplication said to one companion) [Al-Bukhari and Muslim]. Also, you may supplicate with the following Prophetic supplication: "Allaahumma rahmataka arjo fala takilni ila nafsi tarfata 'ayn, wa aslih li sha'ni kullahu, la ilaaha illa ant."
"O Allaah, it is Your Mercy that I hope for, so do not leave me in charge of my affairs even for a blink of an eye and rectify for me all of my affairs. None has the right to be worshipped except You." [Ahmad and Abu Daawood]
Finally, we would like to comment on two matters that were mentioned in the question:
1- Your statement: “…they were supposed to be just friends”, sounds as if this relationship is permissible, rather it is not permissible to establish any kind of relationship with her since she is a marriageable girl to him.
2- Your statement: "…he is ready to swear on our daughter": It is not permissible to swear by other than Allaah and doing so is Shirk (associating partners with Allaah) as reported from the Prophet . For more benefit, please refer to Fatwa 82456.
Allaah Knows best.