My experience with fuel injected engines is limited to flying light aircraft fitted with them. Some of them could be notoriously difficult to start when hot due to vapour locks developing in the fuel system. One solution was to run the fuel pump for a while to purge the system with cooler fuel. Unfortunately a switchable pump is not an option we have on cars.

Not being familiar with your car - is your fuel pump electrically driven? If so, if you turn the ignition on without cranking the starter can you hear the pump run? If so I guess you could try letting it run for a little while before attempting to start.

That doesn't address why it should only happen on short runs and not after long ones. Maybe there's a problem with a thermostat too?