Flowering herbaceous perennials for clay include the daisy family like: Asters, Rudbeckia, Coreopsis
As well as woodland plants like: Phlox, Aconites, Hellebores, Japanese anemone.
Some bedding plants for sunny spots that will thrive in clay soil are: Geraniums, Primulas, Cranesbill, Pulmonaria
Some great plants for clay soil in full sun or light partial shade are: Astilbe, Astrantia, Kniphofia, Solidago,
Don’t forget climbers too. For sun or partial shade: Ivy, Clematis, Honeysuckle
These bulbs are hardy enough to survive clay soils but they can rot in very wet conditions: Narcissi, Snowdrops
Put drainage material like sand in the base of the planting hole to give them a hand.