Here is the final recipe for how I got this thing working.
Hardware: HP Envy 700 PC with (I think) AMD Radeon R7 240; I say “I think” because after everything is installed, clinfo reports AMD Radeon HD 8500 Series.
OS: Ubuntu 18.04.4 with 5.3.0-28 kernel. The kernel is important – using 5.4.0-70 (the default) doesn’t work.
When installing 18.04.4, turn off “download updates while installing”, and once Ubuntu is running, politely decline any updates that are offered. Then install the extra stuff we need:
sudo apt install autoconf libtool libcurl4-openssl-dev pkg-config ocl-icd-opencl-dev
Download the Ubuntu drivers from AMD and install with sudo ./amdgpu-install -y --opencl=pal,legacy
Then get the cgminer 3.7.2 source and:
autoreconf -i
./configure --enable-opencl
make
sudo make install
cgminer -o <pool url> -u <username> -p <password>
And it is off and running! The pool restarts it several times, downgrading the difficulty from 8192 to 1638 to 512, and then once more because it detected a new block (surely could not have done that without my help). Then it kinda sits there with a bunch of zeroes indicating it is contributing precisely nothing. The Slush Pool dashboard hasn’t registered so much as a blip yet, but surely we need to leave this thing running for a while to see anything. So it’s back to bed for me.