The ash should help balance the pH, it is extremely alkaline and can contain stuff that may kill the plants if applied to liberally. It should increase the CEC (cation exchange capacity) of the soil, which you want in the root zone. However, it can also react with the nitrogen in the soil that you want for plant growth early in the season, which defeats the purpose of applying nitrogen.
I have heard of the ash with potatoes as an insecticide, but I thought it was for the beetles. There is a research paper on it which is based off an old farmers technique that discusses application rates. I haven’t tried it. I just saw an experiment literally 30 years ago. You most likely know more about it then I do.
I would apply everything right before fall digging and just mix it in when I was digging to make sure it is getting in the root zone which is where you want it if I was just looking at a soil amendment.