I have the heavy clay soil here in California too.

Dump a gallon of water in the hole, more if you can. Wait a few hours
for it to soften the soil. Gypsum will help soften clay soil, they
also have water additives that help it go into the soil better if you
want to get crazy. When the soil is soft you can go in and work on
the roots, are they too big for a hand saw?

One thing about the clay; it is famous for making those adobe bricks.
So try not to work with it when it is too runny. It's hard to work
with and will harden like a rock!