Mackay was sacked in December after a series of disagreements relating to the Bluebirds' summer transfer business.
The club surprisingly decided to again raise the issue on Thursday by releasing a statement in which chief executive Simon Lim again criticised Mackay, particularly the signing of striker Andreas Cornelius, which Lim claimed cost the club in excess of £8.5million.
Cornelius returned to FC Copenhagen in January and scored a hat-trick on his second debut for the club.
The League Managers' Association chief executive Richard Bevan responded by claiming Mackay had to report to Lim "in respect of all matters pertaining to the transfer of players".
The statement by Lim did however, praise Solskjaer's work in the January transfer window, but the Norwegian did not want to offer his thoughts on Mackay.
"It is early doors for me and it is encouraging they are happy with what I have done but what happened in the past is not for me," he said.
"It guides what resources they give me and what remit I have. I'm just running the club as I would like to.
"It's not got anything to do with me, what happened before. I am just focused on my team because the January window was chaotic."
But Solskjaer did suggest the money spent last summer, when Cardiff broke their transfer record three times, had left him with a smaller budget to work with, and was very positive about his relationship with owner Vincent Tan.
"I have the resources that were given to me, it was different in the summer, they had more money as they had just come up," he said.
"Now it is not like that, it was not right when people said I had £25million to spend, but you have to look at the long-term, you can't just panic and only think of the short term, I only did what I could with the resources I was given."
He added on Tan: "I have spoken with the owner quite a few times, he is very committed to the club, he wants us to be successful, he wants me to be successful. He helps me, he has been everything I hoped for."
On the pitch, Cardiff prepare to face holders Wigan in Saturday's FA Cup fifth-round tie.
The Latics were relegated from the Premier League last season, just days after completing cup success over Manchester City and, with Cardiff currently in the relegation zone, Solskjaer hopes the Welsh club do not follow Wigan's lead.
"They proved last season what the FA Cup was all about, a chance for smaller clubs to win trophies," he said.
"I would like to go one better than they did and win the cup and stay up, but in any way, shape or form, if you play in a cup game you want to win it.
"It's a chance for some players to come in and show they should be playing in the league and I do not go into any game hoping we lose."
Solskjaer was also keen to end his war of words with Swansea head coach Garry Monk following last weekend's south Wales derby.
Solskjaer criticised Jonathan de Guzman's reaction to being hit by Craig Bellamy, who subsequently received a three-match ban, while Monk defended his Dutch midfielder and accused Cardiff's Kim Bo-kyung of punching and headbutting Wayne Routledge.
"I am not aware of it that incident", said Solskjaer. "I would like to forget that game. We got no points and a three-match suspension for Craig, who did nothing.
"I cannot comment on what he (Monk) said, but I did see someone (Angel Rangel) kick Wilfried Zaha off the ball.
"But this rivalry between Cardiff and Swansea and us being in the Premier League is good for us.
"I think we push them and they push us so lets not make it like we are proper enemies and accuse someone of something that has not happened."