I don't know where the relay is, or whether the 500X even has one. Newer cars tend to operate a lot of previously "relayed" circuits via the Body Control Module (aka "Body Computer") to cut down on the amount and cost of components and wiring etc. The BCM is just a PCB behind the dashboard.
They *can* go wrong, which is a whole different level of aggro'... but go with the most "likely" failure points first... which is pressure and sensors.
Your photo is the sensor.... or at least, it looks very similar to the one on my old Stilo. If it's connected to the aluminium a/c tubing then that's the only thing that it could be.
I dunno how you test the pressure sensor but an auto-electrician or a good a/c place (not just the "top up" places) would be able to measure whether it's producing the right voltage when it should... unless someone on here can tell you how to do it..
Even if you have to pay for it, getting it tested will be cheaper than having the a/c emptied, replacing the sensor, re-filling the a/c and then finding that it still doesn't solve the problem. The a/c gas is toxic anyway (I think deliberately venting it into the atmosphere is a "magistrates" kind of offence).... so emptying the a/c to change the sensor is not a DIY job, even if you knew for sure that the sensor was duff.
Ralf S.