Upgrade - personal desktop
 

I bought a pre-built desktop with the following specs in 2014 and now looking to upgrade some of the hardware.

Motherboard: DH87-4440 Asus H87M-E
CPU: Intell Core i5-4440 3.1-3.3GHz HD Graphics 4600
PSU: Min-ATXB Mini Tower ATX Case High Efficiency 80Plus 350W PSU
Hard drive: ST1000DM003 Seagate Barracuda 1TB 3.5" 7200RPM 6Gb/s 64MB SATA
RAM: 4GB 1600 DDR3 RAM x2

I'd like to do some lightweight commercial CFD (CFX/Fluent) and also to test my own code (MPI/CUDA). Hence I'm thinking of the following:

+ GTX1060 6GB
+ 500GB SSD

I wonder if my 350W PSU will be enough for 1060. The official recommendation is 400W but that is based on a system with i7 CPU.

Any comments / advice is welcome.

It depends on the quality of the PSU.
The system will draw much less than 350W, but crappy power supplies tend to turn off before the maximum power on their label is reached. Or even release their magic smoke in very unfortunate cases, taking other hardware with them. Even more so when they are older. That is the main reason why graphics card manufacturers recommend unrealistically high wattage power supplies: to have more headroom for cheap PSUs.
And to be honest, "High Efficiency 80Plus" sounds pretty crappy to me. What brand is it? Does it even have a PCIe power connector for a graphics card?