That power supply that I showed in the first picture was dirt cheap. $16.00 delivered from China. It even has a "trimmer"? that will set the voltage from 11.5 to 14.5 volts. Vendor calls it a" voltage output fine adjustment."
Snub, don't feel guilty, I'm just kidding about working on this particular circuit problem!

Also nothing wrong with your power supply -- I just meant add a very small second power supply -- like a wall wart -- which most people have kicking around from unused equipment.
The second power supply will provide the relay coil with a stable voltage when needed, rather than depending on the output of the PWM to do it, which is a variable voltage.
Since the relay coil doesn't need much power, a small wall wart type power supply of the right voltage and current rating should be easily able to handle the needs of the relay coil. This small supply could even be hard wired and built into whatever case you will be putting your main power supply in (if any).
I actually have a commercial CNC controller that had a big power supply (like yours) for the stepper motors, and a small wall wart to power a circuit board, built inside the aluminum enclosure). I only saw the wall wart when I opened the case of the controller to set some DIP switches for different steppers.
So it really doesn't matter what power supply you have chosen to power the motor, this second small DC supply would only be used to close the relay.
The output of the PWM isn't the best thing to use to operate the relay, since, well, if you turned the control down to zero, the relay couldn't pull in at all. And it may not be able to pull in at speeds higher than that and into your useful range of table travel speeds. So the simple thing to do is not use the PWM output to power the relay.
I suggested a wall wart rather than a diode (and you probably don't know what that is), because you're more likely to have a spare power supply on hand,, and it would be a very simple and straightforward circuit change for you. And easily understandable.