Stoke manager Tony Pulis is planning to appeal against Steven N'Zonzi's sending off during the thrilling 3-3 draw with Southampton at the Britannia Stadium.
The midfielder was shown a straight red card in the 71st minute by referee Mark Clattenburg for a challenge on Jack Cork.
Pulis said: "The disappointing thing is Steven's sending off. Steven has moved his foot across, Cork's on the ground and he has planted his foot in between Cork's legs. I don't think there is any intent there at all for him to hurt the lad.
"I think what might have affected the referee was Cork's reaction. Steven hasn't stamped on him and hasn't touched him with the leg that goes on the ground.
"So that's a disappointment for us and we will certainly be appealing. I am just hoping that Mark will have a look at it and, after some of the incidents that we have been involved with this year, that he will look at it favourably for us as a football club."
Stoke trailed 3-1 after 36 minutes before mounting a second-half fightback with Cameron Jerome's stunning late goal securing a point.
Pulis added: "I thought we were awful first half and really that was as poor as we have been at both ends of the pitch.
"Kenwyne Jones' chance where he misses it early on, it is harder to miss than score and that epitomised everything we did. We were poor defensively, we gave poor goals away and we just looked off the pace. They were still tapping themselves on the back from the Liverpool performance.
"But second half we played with a lot of spirit and a lot of commitment and in the end even with 10 men we are disappointed we have not won the game."
Bury midfielder Steven Schumacher sparked a second-half fightback and the Shakers were unlucky not to grab a share of the points.
Carlisle made a dream start with two early goals. The first of them came in their first attack in the second minute when Brad Potts cleverly tricked Efetobore Sodje on the by-line, forcing the Bury defender to foul him from behind and Lee Miller hammered the subsequent spot kick low into the bottom left-hand corner.
Matty Robson extended Carlisle's lead in the 10th minute, turning adroitly near the penalty spot to drive Frank Simek's cross home left-footed after the full-back had made a raiding run down Bury's exposed right flank.
Ironically Shakers could also have scored twice in a blistering opening spell as Tom Hopper's header was pushed on to the crossbar by Carlisle goalkeeper Mark Gillespie then the on-loan Leicester forward should have done better with a David Worrall cross, which he turned wide from a good position in the box.
Bury came out much-improved after the break and it was no surprise when former Carlisle midfielder Schumacher pulled a goal back in the 53rd minute, planting a firm header past Gillespie after substitute Sebastien Carole had provided the cross from the left.
Gillespie made fine saves from Schumacher, Worrall and Carole as Carlisle clung on to their narrow advantage and again the goalkeeper came to the rescue to deny Hopper in injury time.
The defeat at Oakwell left the Tykes in 23rd place just after the halfway mark of the season with only 21 points from their 24 games.
Director Don Lowing said: "Due to the string of results we've had recently we have dispensed with Keith's services.
"This is a results-based business and we find ourselves at the bottom of the league and unfortunately it has not quite worked for Keith. The hard work he has put in hasn't really got out of it what he has put in so it's a sad day.
"The last home win was 1 September against Bristol City so it's a long time in coming and of course the supporters do get restless and they are looking to us to make a positive move."
Barnsley are now searching for their fourth manager in three and a half years following the departure of former Blackburn defender Hill, who replaced previous boss Mark Robins in June 2011.
