You are talking about black gold that has a value of AU$40 per 25 litre bag in my area. I would suggest either selling it or putting it back into the soil.
See https://en.wikipedia.org/wiki/Biochar for uses of charcoal as a soil amendment.
You can collect (cool) charcoal in a plastic bag until you have a volume worth doing something serious with. Because it's pure carbon it will store indefinitely.
Alternatively, if you rake out the old charcoal, set up your new fire, then put the charcoal on top, then I think you'll find that it does indeed burn. Wood ash tends to be very fluffy and acts as a good insulator — which is why charcoal tends to accumulate in a bed of ash. If you separate the charcoal from the ash then it's no-longer insulated, and getting it to burn is much easier.
Whatever you do, don't pour ash onto a fire. That's about the same as throwing sand onto a fire — something that folks do (in a pinch) to put a fire out.