The Canaries currently occupy 15th in the Premier League table, four points above the drop zone, but West Brom, Crystal Palace and Sunderland below them all have games in hand.
Norwich have won just twice from their last 15 league outings and a disappointing 4-2 defeat at Southampton, in which Hughton made five changes to his side, only added to their mounting fears of relegation.
The 55-year-old's job has been under scrutiny for some time following a big-spending summer, but Snodgrass insists that the players try to ignore any speculation and focus on their game.
"It seems like the manager has been under pressure in the media most of the season," he said.
"From my side as a footballer, you don't think like that. He is the manager, you have your team-mates and you work for that result at the end of the week.
"Against Tottenham we won when people were saying that was a big game, against West Ham we won, against Hull as well. In terms of the unity it is there for all to see.
"Whatever anyone above us thinks is something you can't affect as a player. All I can affect is my performance to help the team win football games."
Snodgrass also believes it is important to give managers time to turn things around, nodding towards West Ham's recent revival as a case-in-point.
He said: "If you look at Sam Allardyce there is the proof of the pudding. It is about faith and loyalty but let's be honest there isn't any loyalty these days.
"You see people lose their job within two months. All Chris Hughton can do is get results."
With eight Premier League games remaining, Norwich face a stern test in the closing stages of the season with Liverpool, Manchester United, Chelsea and Arsenal awaiting in their final four games.
Snodrass knows it will be a tough finale to the campaign and says that the team needs togetherness if they are to secure their top-flight status.
"You're not going to get anything out of a relegation dogfight if you don't stick together so we need to do that as a team," he said.
"It is just not happening away from home right now. If you put performances like that in, it is definitely not going to happen. We let ourselves down playing like that.
"We battled towards the end but we can't give a team like Southampton three goals. It doesn't work like that. I really can't put my finger on it but we will keep going because we're in a relegation fight down there.
"We could go down so we just need to roll our sleeves up and keep going."
Norwich have scored the third-lowest amount of Premier League goals this season, with summer signings Ricky van Wolfswinkel and Gary Hooper yet to find the net in 2014, but Snodgrass refused to lay the blame anywhere.
"We all have to take responsibility for the goals. I would never have a go at the strikers. Their records speak for themselves.
"They all want to score goals, they have that hunger and they would be disappointed themselves, but it will turn.
"You just have to keep supporting the boys. I've scored a few goals and chipped in with some assists and I've had chances to score more.
"This is the happiest I have been this season because I have been hampered with injury from the start."