Hi:
Those ID's confirm that your notebook has the Conexant AC-Link audio chip.
OK...here is what I did. A little detective work...
I like to use the free 7-zip file utility to extract the component files from the exe.
When I extracted the files from the driver that is posted by HP for your model that I linked you to above, here is the hardware ID for the Conexant Audio chip the driver supports.
[Conexant]
%*WDM_AMCAUD.DeviceDesc%=WDM_AMCAUD,PCI\VEN_8086&DEV_266E
As you can see, this driver supports a different Conexant audio chip than the one in your notebook, because the hardware ID you posted is PCI\VEN_1002&DEV_4370
So, I googled the hardware ID and found several drivers that had the main ID of PCI\VEN_1002&DEV_4370, but they had a different &SUBSYS number.
I had to download and extract the files from a few of them before I finally found one that matched the exact hardware ID as yours which is PCI\VEN_1002&DEV_4370&SUBSYS_3091103C
Please see if this one works...