Smith surprisingly quit on Tuesday just nine months after arriving from Australia and helping Trinity avoid relegation from Super League.
Wakefield have just one win from five games this season ahead of tonight’s trip to St Helens and Carter conceded his relationship had turned “frosty” with the 61-year-old since Christmas.
Smith took to Facebook yesterday saying it takes “huge commitment and loads of energy” to run the football side of a club where “costs are cut to the bone.”
He urged fans to “react before it’s too late for a great old club” before signing off with an appeal to ask better questions to get to the “real truth”.
Kirmond said: “I have a lot of respect for Brian. He’s done a lot for us and came in and did his job (avoiding relegation).
“But I can’t see why he’s said that, if I’m honest. Everyone knows the club doesn’t spend as much as others and that it has been in difficult positions before.
“I’m not worried about the future of the club at all and I think what Brian said is probably just a bit of spite towards Michael after he said his piece earlier. It’s a difficult one as I’ve worked with Brian and have respect for him.
“But Michael took this club over when nobody else wanted it and had to cut costs pretty much everywhere to reap back what was lost before in previous regimes.
“This will create a fuss among fans now. It’s the modern way. People jump on the back of things. But I don’t have any worries about Wakefield.”
The second-row admitted he knew the relationship between head coach and chairman had soured.
“It was a shock when Brian came in on Tuesday to say he was leaving,” said Kirmond.
“He said it was nothing to do with us as players and was just personal reasons. But you could see the relationship between them wasn’t as close as when they first got together. They weren’t arguing in front of us or anything – they kept it professional – but you could tell.”
Kirmond had hoped to return from a neck injury for Friday night’s trip to St Helens, but says it will be next Friday at Hull FC when he is now likely to be back in action.
On the visit to Saints, where Trinity have not won since 2009, he added: “We know we have to be on our game to get anything from there.
“Super League’s a tough competition as a whole and it’s not easy getting two points.
“We’ve been putting in performances for 50 or 60 minutes but we’re looking for the whole thing this week.
“Saints’ form has really picked up after a couple of scratchy weeks and having James Roby back firing really helps them.
“But we’re going there to get the win.”