You have to change trains once, in Szolnok, with quite a long hangover:
Cluj Napoca 08.01 - Szolnok 12.45 R364 ADY ENDRE
Szolnok 18.29 - Kosice 23.20 EN578/380 KRAKOVIA
Otherwise, more conveniently, but with a detour via Budapest:
Cluj Napoca 08.01 - Budapest Nyugati 14.10 R364 ADY ENDRE
Budapest Keleti 15.00 - Kosice 18.12 IC536 (in this case mind you that there is a change of stations in Budapest, the Nyugati lies 15-20 minutes walking from the Keleti; otherwise you can take the tramway, 5 minutes transfer)
Best regards and have a nice trip,
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