All that means is your cord is capable of 250V supply, not that it's actually pulling/using any more than your PSUs rated input (110V in NA).
The PSU has no 'toggle' on the rear to switch from 110-240V, so it's adaptive (automatic).
https://www.newegg.ca/Product/Product.aspx?Item=N82E16817171054
I don't think the issue stems from using the "wrong" cable (outside of higher wattage PSUs that need thicker gauge wires, the wall plug cable is fairly generic.
The PSU is far from the best, and if you're pulling anywhere close to it's theoretical max, that may well be the problem.
What are your full system specs?