The Republic of Ireland international will not be back in contention until the new year after sustaining a grade two tear.
News on Robert Snodgrass' knee problem is slightly better, with Hughton saying he will be out for a few weeks but not as long as Pilkington.
"Anthony has had a scan which shows a grade two strain which usually means five or six weeks out," canaries.co.uk reported Hughton telling the club's annual general meeting.
"Robert came back from international duty with Scotland and felt his knee at the beginning of training on Friday.
"He travelled with us to Newcastle as we hoped it would settle down, but unfortunately it didn't, and he could be out for a few weeks but not as long as Anthony."
Striker Ricky van Wolfswinkel (toe) is expected to be fit for the home match against Swansea in mid-December but the comebacks of Mark Bunn (ankle), Alex Tettey (ankle) and Elliott Bennett (cruciate ligament) will all be longer.
Injuries have not helped the Canaries' run of form which has seen them win just once in their last six matches, dropping them to 16th in the Barclays Premier League.
"You go through good periods where you've got no injuries, you keep doing things the same, and then you can pick up some," added Hughton, in comments reported by the Norwich Evening News.
"It's just unfortunate we're going through one of those periods at the moment, but we have to be able to cope with that, and it'll give others an opportunity.
"I'm not going to sit here and say I'm completely happy with our performances.
"Are we where I'd hope we would be at the moment? No, we're not. It's my responsibility to make sure we do something about that, starting with Crystal Palace on Saturday."
Source: PA
Source: PA