If it is the model I found it actually boasts 174W on the 12V rail. So they didn't mislabel it.
Not something I would risk myself. Either it can't handle it or the PSU will live a very short life and burn out. Possibly taking things with it.
Something like a GT1030 or RX550 would be safer and get you better graphics performance than the i5-7400 offers.
If you still wanted the RX560, at least track down the 210W power supply for that system.
If you have enough money, and you plan on doing more PC customizations, it would be a good time to consider replacing the chassis, power supply, and motherboard (And the OS too) You could keep the CPU, memory, and storage. But in some regards there is more value in selling that system as is and...