"U.S. citizens should exercise caution in the regions of Odesa, Kharkiv, Zaporizhzhia and Kherson. In addition, due to a recent increase in low level terrorism incidents, travelers in the cities of Odesa and Kharkiv should exercise extreme vigilance in public places after dark."
"The Foreign and Commonwealth Office (FCO) advise against all but essential travel to:
"Dans ce contexte, il est recommandé de faire preuve en toute occasion d’une vigilance renforcée et d’éviter les rassemblements sur la voie publique."
To sum up it seems it might be ok to visit Kharkiv CITY if you avoid public places after dark and public events.
Anyone is there now to confirm that travel by train to this city for a week-end is relatively safe?