I think its meant as a wood chuck, mine broke and the replacement they sent also went westward soon after but it did give me another face plate..
In my humble opinion the list of jobs to do on the 9x20 in what I think is a good order is...
1, Make a solid plinth to replace the compound, you can thread at 90' so you really only need the compound for short tapers, This mod alone boosts rigidity by a huge percentage and makes parting off a breeze.
https://www.hobby-machinist.com/threads/plinth-for-qctp-on-9-x-20-lathe.45628/
2, Get the QCTP and extra holders of choice, AXA works well with these lathes, I like the wedge type but there does not seem to be much difference in rigidity overall. Make sure the centre post is the correct size for your tool post holder. The one that comes with the lathe is a bit flimsy.
I bored out the original 4 way tool post so it would fit the 14mm shaft required for the QCTP and could be used until you get the qctp
3, A real 4 jaw chuck, dont forget a back plate unless you want to make your own.
You can set the block up in the three jaw by using the three jaws to the side, ie, two jaws on one side and one jaw on the other side and squeezing the block between them, if you lucky the hole will be where it can be bored. (its worth a look to see if it will line up)
The first couple of shots show what I mean by gripping at the sides of the jaws and how the centre would be off set if using the three jaw, you can also off set the same way with the 4 jaw.
https://www.hobby-machinist.com/threads/a-vertical-slide-for-a-9x20-lathe.49724/
also how much stickout can be accomplished with care and fine cuts.
https://www.hobby-machinist.com/threads/the-9x20-mods-thread.30050/
post 12 to see offset boring the compound slide to fit the 14mm post.
Hope this helps.