"I have to hope the fans will become supporters because we need support. We are not deliberately losing games," he said.
But speaking at a scheduled news briefing ahead of Saturday's crucial home match against Bolton, Hodgson suggested his words had been "misinterpreted."
"It was not meant to be an offensive comment in any way," the 63-year-old said.
"I was responding to a question about how it felt to be jeered by fans, making it clear it hurts me and I was disappointed by it because no-one wants to feel they are unpopular.
"I was taking responsibility and prepared to accept that I would be in the firing line. I'm very disappointed if that has been transformed into me attacking the fans.
"I understand it is up to me to take it on the chin but it's not been an easy ride for me. It's been an uphill struggle and I was not the first choice appointment with a lot of fans."
Hodgson, who joined Liverpool from Fulham in July 2010, has presided over Liverpool's worst start to a season in more than half a century.
Hodgson accepts criticism after defeat
The 18-time league champions are 12th in the table with 22 points from 18 games, their lowest total going into the new year since the 1953-54 season, which ended in relegation.
The Reds were also dumped out of the Carling Cup after a home defeat on penalties to League Two side Northampton and slumped to a new low with their first defeat to Wolves in 27 years.
Hodgson said he understood why fans were so unhappy with the team and their manager.
"Fans always have the right to show their anger and disapproval - we've got fantastic fans but you have to give them something to be passionate about and we didn't do that on Wednesday," he added.
"I don't think they were happy with anyone and as manager you are first in the firing line.
"The Anfield support will always be there and what's more it's better in times of crisis."
Several of Friday's newspapers suggested Liverpool owners New England Sports Ventures are running out of patience with Hodgson, with some reports suggesting the former Blackburn boss was on the verge of walking out.
But Hodgson insisted he was determined to ride out the storm and was confident he could bring success to the club if he is given time.
Houllier says Liverpool must back Hodgson
"It's taken me a long while to get to this elevated position coaching one of best clubs in Europe," he said.
"I was really pleased to get the job and left a good job to take it. The last thing in my mind is walking away from a club like this or even walking away from football.
"I want to be here. I want to change things and help the club and the new owners get the success for the club they want.
"I can only beg for patience and the time it will take to do it.
"I'm confident that given time, with the competencies I know I have, the players we have at the club and the possibility of adding to that in future transfer windows, this club will once again be up at the top."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.