Hodgson had to listen to Liverpool fans chanting the name of former player and manager Kenny Dalglish as goals from Ricardo Fuller and Kenwyne Jones ended a six-match unbeaten run in Saturday evening's 2-0 Premier League defeat at the Britannia Stadium.
Asked if the singing of Dalglish's name disappointed him, Hodgson said: "I suppose it does. That seems like it is going to be part of life at Liverpool.
"I think we are trying to do the best job we can in the circumstances and I don't know that that type of singing helps anyone."
Hodgson is adamant it will take time to produce consistent results with his current squad and is determined to fight on in the job he only accepted in the summer.
"If the club decide they want to give the job to somebody else, then I'll have to accept that, if that day comes," said Hodgson
"We are short of players, as everyone knows and we don't have so many options.
"The fans can chant for whoever they want and it will be up to the club to decide what they want to do.
"There is nothing I can do about it and I can only continue to do the best job I can do under these circumstances."
He added: "I can't get upset every time the fans chant someone's name and furthermore, we have millions of fans.
"Maybe the travelling fans are showing their frustration because we've lost, they are entitled to do that.
"Fans make their frustrations felt every time we lose, but unfortunately they may have to do that a few more times, because I can't see us going through a season winning every game."