If your in clay I would suggest going up as opposed to tilling deeper.
Best thing, ADD ORGANIC MATTER!!!
Whatever you have that will break down and be ready for spring can be added, assuming it will not throw off your ph. I would stay away from the branches unless they are smaller than about 1/2" dia, otherwise they could take too long to break down.
Normally I add leaves, manure, and ash -till that in about 6" deep then add a layer of leaves / mulch on top. In the spring I till that top layer in and plant.
Good luck

If I were to do it , I would take all the top good portion out. (pile it on the side)Then dig inside the box , a good shovel depth. Then add compost, About 6 to 8 bags. and mix everything real good. Then return whatever you took out. Add about another 4 to 6 bags of compost/manur and mix it all together. Clay when amended with organic matter is good.
If you do this. you will have more than enough depth for just about any vegetables.
In general, you need a minimum of 10 to 12 inches of good mixture.

With a 4'x4' it should not be very hard to do.