Impossible to index front chain ring! My chain is struggling to get onto the big chain ring and when it does it has quite a bit of chain rub.
The limit screws are quite literally on the limit! And there doesn't appear to be a barrel adjuster.
I tried loosening the cable bolt and pulling on the cable to tighten but it didn't seem to have anymore to give.
Anyone have any advice?
Chain on small cog on rear cassette.
It does look like the cable has missed the channel it should come up through the mechanism vertically, i.e. it is at the yellow arrow but should be at the blue arrow aperture (if I'm identifying the cable correctly):

Little differences like that often mean that the force is incorrect, or that the cable is restraining the movement of the derailleur.  If it's otherwise correctly routed, you might get enough slack another way, to guide it into the open aperture i.e. by moving the derailleur out by hand. Superglue is quite effective at tidying up a frayed cable if you need to re-route it. Twist it so neat and then dab in superglue. Not too much glue tho or it's too bulky.
Double check the support screw is doing work against the plate as well.
Shimano tech doc extract:

If you haven't made use of the Shimano Dealers' Manual then give it a study. Set up on this style is quite different to traditional designs. There is even a mark to show when cable tension is correct and advice on which trim position to use for setting the high limit.
There is a kind of in-buit barrel adjuster, that outward facing Allen bolt on the top assembly. I think you would loosen it off when the cable is undone and once cable is in again, tighten the bolt to add tension.