Not necessarily if the PSU is poor quality. Wattage is often useless without quality. The 80+ certification is only in relation to efficiency.
What actually occurs when you install the new CPU?
Have you got latest drivers such as chipset etc?
Do you also have latest BIOS for that particular board?
Are you able to test another GPU with the new CPU?
It could well be that which such a poor PSU that the newer CPU is causing it to fault. I'd be doubtful, but it's certainly plausible with any bad PSU. Especially being as the GPU seems to work seemingly fine otherwise.
Yeah, sorry. I actually took a look at amperages, which shows a little bit more how efficient the PSU is, and it's actually 38A on the 12V rail, not bad at all. I searched for amperage requirements on GPUs, and this one is much more than enough:
GeForce GTX 1080 - 30A and a 500W PSU minimum
GeForce GTX 1070-26A and a 500W PSU minimum
GeForce GTX 1060 -20A and a 400W PSU minimum
GeForce GTX 1050Ti - 18A and a 350W PSU minimum
GeForce GTX 1050 -17A and a 350W PSU minimum
So we can discard PSU, unless it's very old