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, you should know that it is not permissible for a Muslim man to have an affectionate relationship with a non-Mahram woman outside marriage; for more benefit, please refer to Fatwa 81356.
Therefore, it is not permissible for you to continue this forbidden relationship with that woman whether or not your wife took a promise from you [not to contact her].
Concerning what you mentioned in regard to your wife putting the Mushaf (the Arabic text of the Quran) on your head and taking a promise from you, then if what you meant is that you made an oath to Allaah on what she wanted, like telling her : “I vow to Allaah not to do….” or “I make a vow to Allaah not to do….” then this is an oath and if you break this oath you should expiate for it; for more benefit, please refer to Fatwa 102811.
On the other hand, if you only meant that you promised your wife not to do so, then it is desirable to fulfill this promise but you are not obliged to do so according to the view of the majority of the scholars . Therefore, you are not obliged to fulfill the promise that you made to your wife in regard to not marrying that woman. Putting the Mushaf on your head or the presence of the Mushaf does not affect this ruling.
Hence, it is permissible for you to marry whomever you want [this woman or any other woman] if you are able to be just between your wives in regard to marriage and its attendant rights. However, you are not obliged to seek the permission of your first wife and you are not obliged to inform her about it as we explained in Fataawa 85948 and 84411. But if you feel weak and you predominantly think that this marriage will lead to the break up of your family and to problems that you could not bear, then in this case, it might be better if you refrain from marrying this woman and you should avoid any means that may lead you to commit with her what Allaah has forbidden, so it is not permissible for you to talk to her or be in seclusion with her and the like. For more benefit, please refer to Fatwa 95416.
Allaah Knows best.