Wolves will go into Saturday's clash at in-form Tottenham lying in 16th spot and perilously close to the bottom three.
Relegation rivals QPR this week sacked Neil Warnock while Blackburn boss Steve Kean is under constant pressure from Rovers supporters.
Morgan still has faith in McCarthy but is demanding an improvement in results during the second half of the campaign.
Morgan said: "There is no point hiding. Like my old school report used to say, we could do better.
"Our current league position is not acceptable.
"We didn't want to be in this position again.
"We must improve during the second half of the season.
"It's been disappointing. We expected to be better off than we currently are.
"We all share the disappointment that our fans share with our current points tally and league position.
"I can look back and say we've had some very unfortunate refereeing decisions and bad luck.
"If it's true that it evens itself out over a season, we are going to have some lucky wins in the second half - and we need them.
"But we are not bemoaning our luck. A lot of what has happened has been self-inflicted."
Wolves fans were calling for McCarthy to be sacked earlier in the campaign but Morgan insisted: "We are not a hire-and-fire club."
When asked about Warnock's dismissal, Morgan said: "It's not for me to comment on other clubs. All I can do is comment on Wolves and we do not have a hire-and-fire mentality.
"Of course, we have problems but our philosophy is to work through the problems.
"This club is about stability and approaching things in a very business-like sense.
"Of course, we want the team to do better than it is currently doing. But you don't get long-term growth by chopping and changing all the time."
Morgan added: "The message is finally dropping with people that this is a club about stability.
"You don't get stability by chopping and changing your manager all the time.
"You have to work things through. I'd like to think football thinks Wolves do things our way - and our way is about long-term growth, stability and strength.
"You get strength from team-work."