What is the latest USB host implementation?

Ref. The Arduino USB Host Shield

I'm trying to impersonate a mouse connected to a laptop, and this shield seems the thing to help. But you'll notice that it's retired.

So what's the current recommended hardware and software to do such please?

one possibility : https://www.sparkfun.com/products/9947

A host shield is not for impersonating a mouse; it's for 'impersonating' a PC.

If you want an Arduino to act as a mouse, you can use a board with native USB that can act as a HID; e.g. Leonardo or Micro. You can also hack boards like a Mega or Uno. And there are boards from other manufactures like the Sparkfun ProMicro and PJRC's Teensies that provide native USB.