"It's crazy," Mujica said. "We've got a rough time right now, and I came here to help the bullpen try to get some wins. It's bad to be in this position right now."
Thursday's loss was the 13th in 15 games for the A's.
Mujica fractured the thumb on Bobby Wilson's comebacker in the seventh inning. He also required several stitches for a cut on the thumb.
"It's a minor fracture," Mujica said. "The tendon and everything is fine. Just have to get the swelling down and get the fracture out and start throwing."
The A's are hoping surgery will not be necessary.
Mujica has a 3.32 ERA this season and was acquired from the Boston Red Sox on May 9.
The Associated Press contributed to this report.