Iam not aware of a problem with clay soils, but having said that wood ash is good sprinkled around the base of shrubs. if you get further advice re the clay that says dont mix it then add it to the compost bin.

sorry not much help am I ?

Lightly sprinkling it over the soil is okay, and lightly sprinkling some in the compost heap is okay - but don't add too much because it will make it too alkaline. Don't use excessive amounts on the garden, particularly around acid loving plants.

Because of the alkalinity and the potash content combined, massive quantities can make the clay more dispersed--i.e., more likely to form solid, doughy masses, rather than well aerated crumbs. Massive quantities added to compost can kill some of the beneficial organisms that help work the compost and benefit the soil when the compost is applied.
"Everything in moderation." : )

If the wood ash is collected from a garden fire or a wood-burning stove, you won't have it all year, why not store it in bags in a dry place and use it sparingly around tomatoes and flowering shrubs throughout the year?