There were just two main-draw matches in Russia on Monday, with seeds Kukushkin and Adrian Mannarino progressing.
Kukushkin won the tournament in 2010 for his first and only ATP Tour title to date, and the sixth seed had too much for Jannik Sinner in their opener, advancing 6-3 7-6 (7-4).
Meanwhile, Mannarino, the seventh seed, eased past Stefano Travaglia 7-5 6-2.
At the Moselle Open, however, the one-time winner in action was not able to reach the second round, with Gojowczyk undone by Filip Krajinovic.
Gojowczyk beat Benoit Paire in the 2017 final but went down 7-5 6-4 on Monday.
Fernando Verdasco has struggled for form in recent months - winning twice since Wimbledon - but got the better of Steve Darcis 6-2 6-2, while Frenchman Antoine Hoang defeated Cedrik-Marcel Stebe.
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