The Black Cats travel to in-form Everton on Boxing Day sitting at the foot of the Barclays Premier League table and four points adrift of safety, after passing up an opportunity to close the gap on the teams ahead of them when they could only draw 0-0 with Norwich on Wearside on Saturday.
Poyet admitted after the game that some of what he had seen had been "scary" and that he could not explain why his players had produced such a sub-standard performance.
The Uruguayan's mood will not have been improved by a bout of illness which prevented him from conducting his pre-match press conference on Monday afternoon, but first team coach Charlie Oatway, who deputised, insisted there was no chance of the 46-year-old - or indeed anyone else within the camp - giving up.
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."
Poyet firmly believes the Black Cats are good enough to drag themselves out of the hole into which they have slipped, and nine of the 10 points they have mustered so far this season have come from the 10 league games of which he has taken charge since his appointment as Paolo Di Canio's replacement in October.
However, no-one at the club's Academy of Light training headquarters is under any illusion that a similar return for the remainder of the campaign will be enough to save them from the drop.
Oatway said: "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."
Sunderland will run out at Goodison Park to face a side which is unbeaten in 10 league games and has won four of its last five.
They will do so without the suspended Wes Brown, who was sent off in injury time at the weekend, and with central defensive partner John O'Shea facing a race against time to shake off a shoulder injury, but with their manager having undertaken a forensic examination of what went wrong against the Canaries in a bid to find the way forward.
Oatway said: "In a nice way - and I am, not trying to be funny - Gus is mad, and what I mean by that is he will watch our game two or three times after.
"He will have watched that God knows how many times by now, so he will have digested the pluses and minuses of the game itself.
"For me personally, I just thought we were poor in the first half and in the second half, a bit better, but I still thought we could have won the game.
"But if you don't play well, they are the ones you still need to be trying to win. You need to get results and unfortunately, we didn't get the points.
"But we kept a clean sheet and got a point - but we need to start getting three points."