What should I do in a situation where I am ashamed to be frank with my wife after she discovered my conversations with another girl, knowing that this could lead to separation?

It is not permissible for a man to establish a relationship with a woman who is not his mahram, even if it is for the purpose of hearing beautiful words. This is from the steps of Satan. A woman is forbidden from speaking in a soft and alluring manner to a man who is not permissible for her, as Allah Almighty says: (So do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate speech.) [Surat Al-Ahzab 33:32]. Similarly, it is forbidden for a man to derive pleasure from the voice of a woman who is not his mahram. One must repent to Allah and sever this relationship permanently. There is no need to inform the wife of the details; it is sufficient to state that a mistake was made and has been rectified.