No need to work in the compost. It's necessary to work flour into bread dough and raisins into cookie dough...but the worms and weather will get the compost worked into your soil.
Laying the compost on top of the soil around a plant is called "top dressing" and it works very well......and is a lot less work!
Linda C

And may I suggest that instead of a depression around your trees that you build a small raised wall instead? We typically do this when planting new trees by using the left over backfill. An earth wall arranged in a wide ring around the tree will act like the wall of a moat, holding water over the roots of the trees.

By digging a depression, you may hinder root development into the surrounding soil volume. Believe it or not, the 'feeder roots' of these trees are remarkable shallow, but very wide spread. You want to do as little as possible to disturb them once they become established. A little light cultivating is acceptable, however. 'Scratching' in the compost with a hard rake or long handled cultivator would be a good thing.

If you generally want to work in compost, and it has reached the point where you can't tell what went into it, then you can put it onto soil you've already weeded/dug over and either fork or spade it through the loosened soil. It should cause no harm to new roots.

If it has arrived in bags, or straight from the mushroom growers, then it is safer on the surface. Something makes that pungent smell and its probably a bit too concentrated for young growth.

If it is homemade and still 'identifiable' ie semi-composted, then sieve it first, or take out the big chunks before applying it.

You mention that you've had a dry spell but I'm supposing that you've been keeping your young trees watered. For them you can pile on the compost 4-6" deep, BUT keep it loose so air gets through - and water, and keep it at least four inches away from their trunks.

When it next rains you'll need to loosen the surface of the compost mulch to keep it permeable. A surface cover of short dry twigs will help to do this and stop compost splashing onto your young plants.