You probably want to try a kernel series 4 one.
I do not think you need a series 3 one ,but maybe, maybe not.
If you can get Fossapup to a working desktop.
To change kernel
you can change the kernel to any type you would like with this tool
In a terminal enter
Here are all the normal Puppy kernel packages that can be used:
http://distro.ibiblio.org/puppylinux/huge_kernels/
64 bit kernels have 64 in the package name.
Download a huge kernel tar.bz2 package and extract the files in it to a directory.
open the change_kernels program
open in rox file manager the directory holding the extracted huge kernel components
drag and drop the correct vmlinuz and kernel-modules.sfs or zdrv.sfs paths to the program input slots.
answer prompt yes
The program makes all the needed name changes.
reboot
Should now be using the new kernel.
Just to better understand the change_kernels program.
The info for the input slots is a little miss leading.
(Use nothing from the running Puppy)
What goes into the slots is stuff from the new kernel.
Top slot -->the vmlinuz file
Bottom slot -->the zdrv sfs or the kernel-modules sfs
There is a kernels button bottom right that will take you to the repository of all the regular Puppy versions kernels.
You can use any of them, however get too old a kernel and newer hardware may not be supported.
If you cannot run change_kernels program.
after you extract the new kernel package files.
Change the name of the zdrv.sfs to the exact name it is in Fossapup.
Delete the vmlinuz and the zdrv sfs files from the Fossapup install.
Replace them with the new ones from the new kernel package.
Reboot.