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Will the IMU data update the joint commands received by the hardware interface if it notice the robot don't go straight [..]

The ros_control diff_drive_controller you mean? not automagically, no.

or should i try to fix this in the firmware code?

That, or write a controller / node that compensates for the discrepancy.

If you have encoders I would actually recommend you implement a velocity controlled system: that would make things closed-loop and would automatically take care of any differences between the two motors.