The Black Cats travel to in-form Everton on Boxing Day four points adrift of safety and on the back of a disappointing 0-0 home draw with Norwich City that Poyet described as "scary".
The Uruguayan has been laid low by illness ahead of the festive fixtures, forcing first team coach Charlie Oatway to convey the mood ahead of the trip to Goodison Park.
Asked whether there was a sense of resignation about Sunderland's fate, Oatway said: "No, that won't be happen on our part at all. When it's mathematically impossible, you might adjust and dust yourself down and think about what's going to happen or not going to happen.
"At this moment in time, there's still a lot to play for and a lot of fight in us and in the squad and the club itself, so we will be still trying to march on to get points against Everton.
"But talk is cheap, isn't it? We know there are a few teams around us who are going to struggle or are going to find it hard this season.
"At this moment in time, we are one of them, but we need to make sure that come the end of the season, we are not.
"But like I said, talk is cheap. We just need to start getting some wins. The performances have been getting better and better, even though the last one, I don't think, was one of our better ones.
"But overall, I think it's been improving. I watched the games before we got here and I can see an improvement in the way that we play, the way that we defend.
"But we just need to start winning games now."