The CIA is an unusual suggestion, these have nothing to do with joystick/mouse movement input in any Amiga model.
There are 4 movement inputs (up, down left, right) that are read by U4 (Denise) in the A600, and these four inputs are multiplexed to eight inputs (for both ports) by U34. If an input isn't working, it's either a fault with U34, or more likely, the input signal isn't getting to U34.
It's simple to fault find. Use a DC voltmeter to measure between ground and U34 pin 11 (joystick up) and pin 10 (joystick left). They'll be at 5V with no input, and at near 0V when the input is activated. If those inputs of U34 are changing but there's no movement reported, it's a fault with U34, which needs to be replaced; about 30 seconds work with a SMD rework station.
If the input signal isn't getting to U34, it can be a problem with corroded pins in the joystick connector, cracked inductors (E365, E367 in this case) or possibly PCB damage, all of which you can pin point using the voltmeter to see where the input signal is not getting to. With the computer powered off, you can also measure continuity between joystick connector pins 1 to 4, which should connect to U34 pins 11, 14, 10, 13 via 68 Ohm series resistors.
It's also worth checking the solder joints on the 9-way connector.
In the case of cracked inductors, which sometimes happens, you can just remove them and bridge the PCB pads with a piece of wire. In this part of the circuit they're only there to reduce conducted radio frequency emissions as part of EMC compliance. The input will function exacty the same with a wire bridge in place of the inductor.
