Is the marriage of a girl who had hugs and kisses with her fiancé before marriage considered an expiation for her sin, and what should her husband do to alleviate her suffering after her death?

There is no text stating that marriage expiates sins. Rather, the expiation of sins and the purification of a servant's faith from its impurities in this world occur through four things: repentance, seeking forgiveness, good deeds that wipe away sins, and calamities that expiate sins.

If these four are not sufficient for purification, the servant is purified in the Barzakh (intermediate realm) by three things: the prayers of the believers for him and their seeking forgiveness for him, the trials of the grave, and the gifts of deeds that his Muslim brothers dedicate to him.

If these are not sufficient for purification, the servant is purified before his Lord at the Standing (on the Day of Judgment) by four things: the terrors of the Day of Resurrection, the intensity of the Standing, the intercession of intercessors, and the forgiveness of Allah, the Mighty and Exalted.

If all of these are not sufficient for purification, then entry into the Fire is necessary to be a purification and cleansing for his impurities, after which he will exit it and enter Paradise.

It is legislated to supplicate for the deceased woman, seek forgiveness for her, and give charity on her behalf.