To run the PS4's library of games it would need an x86 processor right?

Nope. I suggest you look into the work Transmeta were doing.
Following that... Intel with Medfield.
And of course we can't forget nVidia's Denver.

I doubt it, Sony is probably done with handhelds

2400$ for a "handheld"console? not sold!

It's technically possible, but just look at that pricetag. And you still would need to compress that into a handheld format. Which will be very tough because of the space needed both for the batteries and the heatsink (which will have to be huge, considering it's TDP of 65W and the fact in a console configuration it will basically always draw it's full power)

If that is a bit too expensive, try this one.
http://www.dell.com/en-us/shop/dell-laptops/new-xps-15-2-n-1/spd/xps-15-9575-2-in-1-laptop/fncwlf417s
~1,300USD, and it still has ~2.5TFLOPs of compute on its main GPU.

Price, size and power draw could all be brought down by removing unneeded components.
Do you really need two GPUs.
https://www.notebookcheck.net/Intel-HD-Graphics-630.187948.0.html
The iGPU (the GPU attached to the CPU) looks to be roughly as powerful as a docked Switch, an easy target for removal in a potential handheld.
Also, the CPU is still massively overpowered for the task at hand.

I never said it was a good idea, or would result in a good product (consoles are budget systems).
The Dell laptop can be configured down to a 1300 USD price-point, retaining two GPUs (one with ~2.5TFLOPs), and a high speed Intel CPU.

You would probably only need a GPU of around 800-1000Gflop to match the Playstation 4.
Provided it has sufficient texture mapping, geometry and render output pipes.

AMD's Ryzen APU's with a 12-25w TDP would fit the bill, the caveat then becomes bandwidth, which the APU's obviously lack.
GDDR5 tends to be fairly power demanding.

I don't think Sony will enter handheld market ever again. They are doing just fine with PS4 and will continue to focus on home consoles. I would say that even Microsoft coming up with some handheld device which plays Xbox games is more likely at this point than Sony.

O....kay....? https://www.gamesradar.com/biggest-ps4-install-size-games/

even if they do. im still salty from the shit support for vita. as far as anything from sony besides the playstation, im out.