One banner in the crowd urged owner Tony Fernandes to replace the Welshman with ex-Spurs manager Harry Redknapp.
"I don't run away from challenges and this is a huge challenge," said Hughes.
Waiting for a win
QPR are the only team in the four tiers of English professional football yet to win a league match this season
"It was a huge challenge when I came here and there weren't many queuing up when I decided this was the right job for me."
After the game,
Fernandes wrote on Twitter:
"I feel gutted. I have put my heart and soul into this with my other shareholders and done all we can to give support to players and all management. I can only apologise to the QPR fans. We keep fighting."
Hughes said he had not spoken to Fernandes or any members of the QPR hierarchy, who have been steadfast in their public support for the manager they appointed in January, since the defeat.
He added: "The fans are well within their rights to criticise me, the staff who prepare the team and the team after that. It was unacceptable. We have to take it on the chin and ride with it, but I don't run away from things and it is my intention to see it through.
"We are going through a lot of pain at the moment and it will only change if we win football matches. It is proving very difficult for us but I still have confidence in the group."
The meeting of QPR and Southampton had been labelled "El Sackico", with speculation that the losing manager would be in more danger of losing his job.
Hughes said: "The club needs stability and that is what we are trying to bring. It was the worst performance since I came here. Maybe it is a watershed in that possibly we can't get any lower than this."
Southampton manager Adkins, in contrast, was elated as pressure eased on him after his team's first away win of the season moved them to within a point of Reading, who are one place above the relegation zone.
"We showed a growing maturity," he said. "Every game in this league brings its own pressures and demands. We have come a long way in a short space of time.
"It is my first year in this league and I understand how the media analyses things. I sympathise with Mark. You are always going to have the intensity of it whether you are at the top or bottom of the table.
"In these early games we have played eight of the top 10 in the league, so for a club to get promoted, or any new team, it was always going to be a challenge, but we have now got our first away win and the players are growing in confidence."
Unlike Hughes, Adkins enjoyed plenty of backing from the stands, with Saints supporters repeatedly chanting their manager's name.
"The fans lifted the players all the way through the game," added Adkins. "We are all supporting each other. Everyone is together at our football club."
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