Didn't work out. A pity. Machine spontaneously rebooted once and its output has errors - so back to how it was with two CPUs and a further test to ensure it is stable again.
Maybe with a reduction in frequency or raising the CPU voltage it might work - but its not worth the time and effort. Believe this Phenom is an X2 that failed tests for X3 or X4. If I really want four CPUs would be better getting a second-hand pull from China such as an AMD Phenom II X4 955 for about $24 US or less.

Well, on the desktop I have just been flashing/updating the BIOS to the latest version dated 01/13/2021.
After that the system rebooted, BOINC was automatically started as always, but I had to update BOINC, too.
Did that, so BOINC was reloaded/restarted again, did its benchmarks and now a fresh new BOINC is running.

The logs say:

Thu 01 Apr 2021 19:52:40 CEST |  | Starting BOINC client version 7.16.11 for x86_64-pc-linux-gnu
Thu 01 Apr 2021 19:52:40 CEST |  | log flags: file_xfer, sched_ops, task, coproc_debug
Thu 01 Apr 2021 19:52:40 CEST |  | Libraries: libcurl/7.71.1 OpenSSL/1.1.1k-fips zlib/1.2.11 brotli/1.0.9 libidn2/2.3.0 libpsl/0.21.1 (+libidn2/2.3.0) libssh/0.9.5/openssl/zlib nghttp2/1.43.0
Thu 01 Apr 2021 19:52:40 CEST |  | Data directory: /var/lib/boinc
Thu 01 Apr 2021 19:52:40 CEST |  | [coproc] launching child process at /usr/bin/boinc
Thu 01 Apr 2021 19:52:40 CEST |  | [coproc] with data directory /var/lib/boinc
Thu 01 Apr 2021 19:52:40 CEST |  | [coproc] NVIDIA: libcuda.so: cannot open shared object file: No such file or directory
Thu 01 Apr 2021 19:52:40 CEST |  | [coproc] ATI: libaticalrt.so: cannot open shared object file: No such file or directory
Thu 01 Apr 2021 19:52:40 CEST |  | [coproc] OpenCL library present but no OpenCL-capable devices found
Thu 01 Apr 2021 19:52:40 CEST |  | No usable GPUs found
Thu 01 Apr 2021 19:52:40 CEST |  | Version change (7.16.6 -> 7.16.11)
Thu 01 Apr 2021 19:52:40 CEST |  | libc: GNU libc version 2.32

Nothing has changed, still getting:

# clinfo -l
cl_get_gt_device(): error, unknown device: 3e92
Platform #0: Portable Computing Language
 `-- Device #0: pthread-Intel(R) Core(TM) i7-8700 CPU @ 3.20GHz
Platform #1: Intel Gen OCL Driver
Platform #2: AMD Accelerated Parallel Processing
Platform #3: Clover

Retrying, moving one file one level up:

# cd /etc/OpenCL/vendors/ && mv intel-beignet.icd .. && clinfo -l
Platform #0: Portable Computing Language
 `-- Device #0: pthread-Intel(R) Core(TM) i7-8700 CPU @ 3.20GHz
Platform #1: AMD Accelerated Parallel Processing
Platform #2: Clover

I don't know if the "unknown device: 3e92" message matters anything.

Update: here is the complete output of 'clinfo' on my i7-8700 desktop, without the Intel Gen OCL Driver that is causing the "unknown device: 3e92" error: output of 'clinfo' for i7-8700 desktop.