The St Andrew's boss, and former Black Cats midfielder, was left frustrated by Sunderland's opener.
Jordi Gomez's struck with just 13 minutes left but Birmingham felt the ball had gone out during the build up.
Last season's beaten finalists scored twice more through Adam Johnson and Connor Wickham in the final three minutes to add some gloss to a unconvincing performance.
A frustrated Clark said: "I'm not speaking about the officials, I'd rather spend my hard-earned money on something else rather than a fine.
"I am going to go through the right channels, I will have some footage for them.
"It's not a true reflection of the game
We played well, the scoreline does not do my players justice
After the first goal the players started chasing the equaliser and Premier League players will pick you off.
"We were more than a match for a very good team.
"I have said to the players, they have played really well
I'm delighted with everyone and they should come to the training ground smiling."
Birmingham had the better of a forgettable first half as David Cotterill hit the post and Wes Thomas fired wide.
But they failed to break Sunderland down and Gomez struck from 25-yards to begin the goal rush.
Clayton Donaldson shot straight at Costel Pantilimon before Johnson tapped in after Jozy Altidore's shot was saved.
Wickham made it 3-0 on 89 minutes when he found the corner from 15 yards to clinch a third-round tie at home to Stoke.
Boss Gus Poyet said: "It was a strange game, it was a poor first half but somehow we got to half time 0-0 but the second half was different.
"The same group of players got something at half-time and the first goal was key.
"It's not going to be perfect all the time, after we scored it was the best we have played this season.
"We played rubbish for 45 minutes but we finished the game on a high
It's the way the cup goes."
Source : PA