The 45-year-old Uruguayan was able to work with his full squad for the first time on Thursday following the return of the bulk of his international players, but spent much of his first few days on Wearside assessing the problems which have left the club marooned at the foot of the Barclays Premier League table.
And the root of the trouble was not difficult to find as he prepared to send his team out for the first time at Swansea on Saturday.
Poyet said with a smile: "To be honest, we have got two problems: one, we concede too many, two, we don't score too many. It is simple.
"I was saying this morning, it's like scales - we need to find a balance and try not to concede too many and score a few more. It is simple in here, but it is difficult on the pitch.
"The idea is to make sure we do the right things at the back, we are organised and difficult to beat and then start playing our game with the quality players where they can hurt the opposition or where the opposition is weak.
"There are plenty of things we need to improve, but slowly. We cannot change everything in 10 days, or especially in two days with the international players."
Poyet, of course, inherited the Premier League's bottom club after Paolo Di Canio's departure just 13 games into his reign, and he did so with the team in some disarray.
They have not yet won a league game this season and have collected only one of the 21 points they have contested to date.
That return is the result of a goals for column which currently stands at just five, some 11 fewer than they have conceded, and Poyet's assessment that they need to work on what is happening at both ends of the pitch is hardly the stuff of astronautical engineering.
However, the prospective return of striker Steven Fletcher from a shoulder injury ahead of schedule could pave the way for a significant improvement in at least one respect.
Scotland international Fletcher has scored 12 goals in 32 league appearances for the Black Cats, and his readiness to play through the pain barrier has impressed Poyet.
The manager said: "The problem was about contact and the other part of football. He was probably fitter than anyone because he has been training on his own, running up and down and doing everything, in the gym more than any other player.
"The idea was to check the whole week this week that he was able to compete and to use his body in the proper way, especially when you play in the position that he plays.
"Everything has been fine until today and I hope tomorrow nobody does anything silly against him and he will be ready on Saturday.
"But it's good news, it's very good news. It's very important that he is desperate to help as well.
"He puts himself about, and sometimes you need to play with a little bit of pain as well. You cannot play professional football pain-free, it's impossible."
Only Jozy Altidore, Ji Dong-won and Ki Sung-yeung, who is ineligible for the game at the Liberty Stadium under the terms of his loan agreement with the Swans, were still to return from international duty on Thursday afternoon, while long-term absentees Phil Bardsley and Wes Brown are also back in contention.