Marlon King, who is one the Blues' highest earners, grabbed a last-gasp winner but is one of a host of players being linked with a move away from St Andrew's.
The victory helped Birmingham move away from the drop zone but, with finances tight Clark admits he could be bidding farewell to a number of the side that featured at Turf Moor ahead of transfer deadline day.
"We need to do something in terms of players outgoing to keep us going," said Clark. "For a couple of days it will be a bit tough, but come Thursday night I will know what I am doing.
"The toughest thing for us at the moment is that I cannot go after the targets that I want. I can't make calls to players and their agents at this moment in time because I don't know what I have got to work with.
"This has been a massive learning curve for me. All the coaching courses I have been on could never teach me how to handle the situation. But I am enjoying it, I am at a massive football club and we want to get it back to where it belongs. I am proud to be the manager here."
Curtis Davies had given the relegation-threatened visitors the lead before Ross Wallace equalised from the penalty spot at the start of the second half. Dean Marney then saw red for the hosts and Birmingham made the extra man count thanks to King's late strike.
Burnley boss Sean Dyche said: "Some of the football was excellent but you are having to earn wins not just by beating 11, we've having to go beyond that. The officials should be there to manage the game and not make decisions that change it radically.
"Apparently [Marney's red card] was for excessive force so I think you've got to tackle softly now. I don't think you're allowed to tackle the ball. Maybe you are not allowed to use your feet in a tackle
"You cannot take a great tackle out of the game because it will kill football. All managers want to play the beautiful game, you all want to be a version of Barcelona, Manchester United, Chelsea. But you can't take away tackling. It's what fans buzz off."