Is it permissible to continue with the second wife, to whom one was married secretly without a guardian, but with witnesses and someone acting on behalf of a notary public? And is the previous relationship with her considered adultery if it is not permissible, and how does one purify oneself from that?

There is a disagreement between the majority of scholars and Abu Hanifa regarding marriage without a guardian (wali). The fatwa (religious edict) is that a guardian is obligatory, due to the strength of its proofs. However, if the marriage contract was concluded without a guardian, in imitation of Abu Hanifa's view or if a judge ruled it valid, then it is considered binding. If it is possible to renew the contract with the permission of her guardian, that is preferable. Otherwise, there is no harm in cohabiting with her, and justice must be maintained between her and the first wife. It is also advised to strive to mend the relationship with the first wife in order to preserve the family. If she refuses to share the bed, then she is considered disobedient (nashiz).