I am in the same situation after a large fire earlier in the year although I have bagged the ash for the moment.
I am also continuing top bag the ash from the fire in the house (we only burn wood at the moment).
I would say that you have 3 options:
1st - just add the ash into a compost heap.
2nd - put round the base of any fruit bushes, trees in the spring to help with flowering and then with fruiting.
3rd - apply to where your spuds will be grown next year - they like a lot of potash.
To improve the soil structure think about adding in compost/leaf mould. Your local authority may have compost for sale.
Grow a green manure in the spring and dig into the soil to help improve the organic content things like - phacelia, borage, vetches, mustards, rye grass, clovers. The legumes will leave nitrogen and also some of them have deep tap roots to help break up any compaction layers below.
We have clay soil although we haven't gotten round yet to grow anything this year but I'm planning on getting the veg beds done over the next few weeks. I'll be adding in a load of bought in compost I think and will eventually get round to doing green manures.