The combo of clay soil, shade and tree roots will make the choice of plant tricky.

Depending u[pon how much time and effort you want to put in I see the issue as 1) construct raised planters or use containers. 2) amend the clay soil.

Using containers for shots of colour is a good way to avoid the clay soil and the tree roots. Begonias and Impatiens are my best pick of annuals for shade areas.

I found a good article of the subject for you:

https://www.gardeningknowhow.com/special/shade/plants-for-shady-clay-locations.htm

I do not agree with some of their plant suggestions. For an area which has a lot of trees, and thus. roots, I would be selecting plants with shallow roots which can deal with the shade and the soil.

Try PennyHapenny (Lysmachia numaleria), Corydalis, Deadnettle (Lamium), HerbRobert (Geranium robertium), SweetWoodruff, (Galium odoratum) Ajuga. These plants are all perennial, and all flower. The Corydalis flowers on and off from early summer to late fall, as does the Lamium.

Another option is some of the ornamental sedge grasses, also with shallow root systems, but usually with interesting folliage.