The Spain international suggested United's 2-1 Premier League loss last weekend was due in part to a worse attitude than David Wagner's side.
Herrera's explanation for their first defeat of the season prompted a furious response from Mourinho, who said: "Oh my God. When a player says that, when a player feels that, I think they should all go to the press conference and explain why because I cannot explain that."
The former Athletic Bilbao man has only made three starts in the Premier League this season and speculation of a soured relationship with Mourinho was exacerbated by the Huddersfield remarks.
However, Herrera insists he still has a "fantastic" understanding with Mourinho and has no problem with any critical comments.
"If you listen properly to what I said, I just said in the first 30 minutes they had a better attitude than us," he said following the 2-0 EFL Cup win at Swansea City on Tuesday.
"That is what happened. After that we had a good attitude and we were even close to drawing the game, which is not easy after playing bad for the first 30 minutes. That is what I said but now we want to forget what happened.
"Everything he says I accept. I am a professional. We have a fantastic relationship, so let's look forward and think about Tottenham."
United's attention now turns towards the visit of Tottenham on Saturday and Herrera admits it is a must-win game against a team with whom they are level on 20 points in the table.
"Tottenham is not going to be four or five points but we have to win against them because they are a direct rival for the title and also because we play at home," he said.
"For the last two or three seasons, they have improved a lot. They are contenders for everything."