From "thespruce.com": (since their references are 2 University Cooperative Extensions, I have confidence in what they are conveying, but then I am a Cooperative Extension Master Gardener so I might be biased
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Is Wood Ash Good for Garden Soil?
Wood ash can pile up during a cold winter and it would be nice to have a practical use for it. It has often been used as a soil amendment in gardens. However in recent years, gardeners have been given mixed signals about the safety and value of using wood ash on their garden soil. Is it safe to use wood ash in the garden?
As with most gardening questions, the answer is "That depends." You really have to know a little about both your soil and the ashes themselves.
What Does Wood Ash Do for Soil?
Ashes from fireplaces and wood burning stoves can be a good source of potassium. To a lesser degree, they also provide some phosphorus, a bit of aluminum, magnesium, and sodium, and a few micro-nutrients, such as boron, copper, molybdenum, sulfur, and zinc. The amount of nutrients in the wood ash is not particularly high and it really depends on the type of wood burned. But if your soil has a potassium deficiency, wood ashes can be a good amendment.
Wood Ash and Soil pH
Most wood ash contains a good percentage, about 25%, calcium carbonate, an ingredient in garden lime. If your soil is very acidic (5.5 or lower), amending with wood ash can raise your soil pH.
On the other hand, if your soil is neutral or alkaline to begin with, adding wood ash could raise the pH high enough to interfere with plants ability to take in nutrients. Wood ash should also be avoided around acid loving plants like rhododendrons and blueberries.
What are the Negative Effects of Using Wood Ash in the Garden?
Unfortunately, wood ash can also be a source of heavy metals such as cadmium, chromium, or lead, that you don't necessarily want in your garden. However most studies have not shown that if the soil pH is above 6.0, the heavy metals are not taken in by the plants in measurable amounts.
And since wood ash raises the soil pH, the presence of heavy metals should not be a problem. If you have a regular source of wood ash and are worried about extended use of it in your garden, you should consider having it tested as a lab.
The bottom line is that a small amount of wood ash will add some nutrients and be beneficial to most soils. Large amounts should be avoided.
Important Note: Never use wood ash in the garden, especially around edible plants, if any type of fuel or wood preservative was used on the wood.
Wood Ash Bonus
Slugs and snail are repelled by wood ash, and if they do come in contact with it, it acts much like salt and desiccates their bodies.
Wood ash can also be used to smother aphids. Dust a fine layer onto infested plants, coating the aphids. You can hose the ash off the plants, once it has done its job.
Sources:
University of Maine Cooperative Extension
University of Georgia Cooperative Extension