Henry Avagyan was transferred to FC Pyunik from FC BKMA Yerevan in 2023. Before the transfer, Avagyan has spent his entire professional career in Armenia. With FC Alashkert Henry became an Armenian Cup winner and an Armenian Super Cup winner of the season 2018/19. Later during the season 2021/22 Henry again became an Armenian Cup winner but this time as a player of Noravank SC. Henry was called up to Armenia national under-17, under-19, and under-21 teams. In 2018 Henry played his first game for Armenia national team in the 2018-19 UEFA Nations League D match against Liechtenstein.