But, after beating Harlequins 23-16 to kick-start their Pool Four campaign, full-back Byrne says he is motivated to finish the season as a winner in Europe's premier tournament.
"This season the boys have come back fresh," said Byrne.
"It's an important competition and we want to put it right. We don't know how long this tournament is going to be around.
"This year we want to come out and win it."
Clermont now face a crunch double-header with Byrne's former side the Scarlets in December.
And Byrne, who scored the opener against Quins, claims his team-mates must start converting their chances into tries if they want to reach the quarter-finals.
"We have Scarlets back-to-back and they are going well," added Byrne.
"It will be a big couple of weeks for us to come out of the group.
"We had opportunities in the 22 and it's been a bit of a problem this season. We've not quite finishing our chances.
"We were pretty disappointed in the dressing room after the Quins game because we didn't get a bonus point.
"So we have to address that going into the next rounds because it's going to be tough."