A coffee maker?
What you do depends very much on what is on the other side of the PCB, if it only has switches and LEDs on it the other components shouldn't be there.
I would tackle the repair in one of two ways, either scratch off the green paint (solder resist) at each side of the damage then solder a wire bridge across the gap -or- use a fine insulated wire to bridge the broken connection but soldered to the start and end points of the track where there are bigger pads to connect to.
As for wiring relays, as long as you are not likely to press a button at the same time as operating a relay, wire the 'common' pin of the relay to one side of the actual switch on the board and the 'normally closed' pin to the other side of the switch. operating the relay will than have the same effect as pressing the switch. To drive a relay from a RPi, you will need a driver transistor to boost the available current from the Pi's output pin, a resistor to limit the current and a diode to go across the relay coil.
Brian.