The South Korean midfielder - on loan from Swansea City - was left out of Wednesday's 2-2 draw at Manchester City and his absence will come as a huge blow to the club's hopes of Premier League survival.
However, Poyet hopes to have the 25-year-old back sooner rather than later, with Sunderland six points adrift of safety at the foot of the table.
"Nothing serious, but it was getting worse, so we had a good chat, he had an MRI, we talked to the doctor in the national team and it was time to stop," he said.
Fernandinho gave City the lead inside two minutes at the Etihad Stadium before Connor Wickham threatened to completely derail the home side's title hopes by scoring twice in 10 second-half minutes.
However, Samir Nasri's late drive was fumbled by Vito Mannone before trickling over the line, with Poyet quick to defend his goalkeeper.
"That was the hardest part because he's been outstanding," he continued.
"He feels terrible now. It's about feelings and it's about controlling emotions.
"We win all together, we celebrate. When we beat Man United (on penalties to reach League Cup final) all together we were all responsible - him a little bit more.
"That doesn't mean I changed my mind. He needs to recover quick. We need him for Saturday (away at Chelsea)."
