I bought it with The GPU, but The Company sell The same model with different hardware inside.
This is my PSU
https://ibb.co/Gs5D0HC
That's what I wanted the picture for because some OEM compoters come with half decent PSUs and most of them come with fire hazards like that; well to cut costs for the company and increase profit margins. A good quality PSU with 320W on two 12V rails might be able to pull it off with a non-OC 4770 and a GTX 1060. Not this one and I'd say you've been lucky so far.
However, that PSU's OEM is AcBel Polytech which is known for making horrible quality PSUs. I would not recommend powering and running that setup even without the graphics card with that PSU.
Actually any system with PSUs from that manufacturer is at constant risk because when low-quality bad PSU like those fail they blow other components with them and first victims are usually motherboards and/or graphics cards.
As you say it's usually not replaceable because those PSUs come with proprietary connectors for motherboard/CPU and if replaced with PSU's with standard ATX connectors sometimes the user can use adapters to connect the PSU to motherboard etc and sometimes it can not be done.
I would suggest that you use the CPU RAM GPU and storage devices and get a compatible motherboard (Haswell compatible chipsets are B85 Q85 H81 Q87 H87 Z87 H97 Z97; although at this stage you'd probably have to get a used one and with a non-K CPU Z-series boards are kind of pointless).
You'd need a good-quality 450W PSU, or to have some head room and possible future upgrades it's won't hurt to get a good-quality solid 550W unit; those come with 7-10 year warranty.