In terms of how to buy compost, you can go to your local garden center, a compost/soil/mulch. When buying compost, it should smell good, not like garbage.
Check with a real garden supply store. They should be able to buy it by the cubic yard and get the really premium stuff.
If anything tends to be 'good enough' because you already bought it or it's all you could find at the very first store you visited (and your idea of 'good compost' .
It's best not to mix ash in the soil where your potato plants will grow, as it can promote. Is it safe to plant vegetables in pots with store–bought compost and cow .