Having played out a 1-1 draw in the initial encounter, the two sides met in a replay on Tuesday and Rene Meulensteen's charges cruised to a 3-0 triumph to set up a fourth-round encounter with Sheffield United.
That extended Norwich's winless run to eight matches in all competitions, and Hughton's position has come under scrutiny as his side sit 15th in the Premier League, just two points above the relegation zone.
The 55-year-old admits his side's poor form is cause for concern, but the former Newcastle United boss is confident the tide will soon turn.
"I understand all of the pressure that comes with this," he said.
"We as a club wanted to go through in this competition, and we had a support that travelled a long way and what they saw wasn't good enough.
"It is a period where we have to make sure we stick together, show strength as a team, because a team going out there must believe they can win.
"My only concerns are doing the job to the best of my ability and getting results for the team. Do I worry about not getting results? Yes, because that's the business we're in.
"But we're also three points away from 10th place and there are teams that have deemed to have had good seasons that are perhaps one or two points ahead of us in the division.
"So sometimes we have to look at what the reality is, but at the moment we're hurting because we haven't done well enough to win
"It's a tough period, but I wouldn't say it's my toughest period. You have to remember we finished 11th last season, but with three games to go we were one of seven or eight teams that could have gone down.
"We've had tough periods, but you have to come out of these stronger and the only thing on my mind is firstly the disappointment of going out, and we've got a massive game on Saturday (against Hull City) we've got to try to win."
