The Black Cats dramatically survived relegation from the Premier League in May following a strong run of form at the business end of the campaign, while they have begun this term with one win from their first seven games.
Having worked on getting his squad to come together as one, Poyet is confident that positive results will soon follow.
"I think the minimum that Sunderland fans want from their players and manager is that it is obvious they care," he told The Northern Echo. "That is something we have had to dramatically improve from last season.
"At the beginning of last year, there was a doubt. When I first got here and we lost 4-0 at Swansea, there was a doubt about how much we were suffering as a group. We beat Newcastle and that helped to change the momentum, and now I think you can definitely see that the team is always trying.
"No matter what else is happening, we are always trying really hard to win the game, or at least not to lose it. That is something the fans in the North-East really appreciate. It is not enough – you need to do that and then also win matches – but it is an important starting point whenever you go onto the field."
Sunderland are currently 13th in the Premier League table.