After some more digging I actually found a solution to my issue in an askubuntu forum answer. The issue as was pointed out by @petsam, was that the incorrect audio driver was loaded.
The way to solve it for me was to perform the following:
sudo nvim /etc/default/grub
Change
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash ..."
to
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash snd_hda_intel.dmic_detect=0 ..."
and then update grub with:
sudo grub-mkconfig -o /boot/grub/grub.cfg
and reboot.
As I understand it, the stock sof-audio-pci still don't have any driver support and thus pulseaudio was not detecting my soundcard. By performing the commands above, you redict grub to use the snd_hda_intel drivers that work find as indicated in HP Pavilion 15-cx0xxx - ArchWiki.
Thanks everyone for the help.