They are being hunted by Donald Pierce (Boyd Holbrook), a cyborg with a heavily armed gang backing him up.
We know that Logan is protecting her on some kind of journey, though the reasons and the destination are still unknown.
As to what could leave Logan in such bad shape that he would need to do some self-operating, that probably has to do Laura — otherwise known as X-23.
Logan’s never been very good at getting out of a fight, so taking on the super powered Pierce and his gang with a diminished healing factor presumably won’t turn out very well for the former X-Man.