The team, the second most successful in the sport's history after Ferrari, are last in the constructors' championship and scored their first points of the season only eight days ago in Azerbaijan thanks to Spaniard Fernando Alonso.
McLaren tested the new Spec Three power unit in Friday practise in Baku and Honda's F1 head Yusuke Hasegawa said they would be racing with it in Spielberg.
"We were able to confirm some progress in terms of the power output," he said in a team preview of the race.
"We have proceeded with the mapping back at the factory utilising the data we collected, and we are going to bring the Spec Three for both drivers this weekend."
Honda's failure to introduce the upgrade two races previously in Canada triggered highly critical comments from McLaren management and fuelled speculation that the partnership was heading for divorce.
McLaren have not won a race since 2012 and have been plagued by reliability problems and a lack of engine performance since they got back together with Honda in 2015.
"I'm hopeful that we'll be in a position to get amid the other midfield runners and take a useful step forwards," said Racing Director Eric Boullier.
(Reporting by Alan Baldwin, editing by Ed Osmond)
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