I've heard of disappearing cursors but not anything like you are experiencing.
Best I can do is show you a thread. from the last post.
PostPosted: Wed 03 Sep 2008, 04:41 Post subject: mouse pointer
Subject description: mouse pointer disappearing Reply with quote
It could be a problem with the video card. I got this fix from Richard Pace:
"The workaround is to disable hardware cursor in xorg.conf.
Add the following line in the device section for your video card:
Option "HWCursor" "off"
Let me know if it works!
from a 2008 thread
Also. After installing a 188.8.131.52 Liquorix Kernel in my AntiX 8.5 instsll I lost tap on my Asus 701SD. I don't know if it will helo but I will post what I did yo fix it any ways. I fixed it by fooling my /etc/modprobe.d into thinking my touchpad was a ps2 mouse. Can't guarantee this fix will work for your problem in Puppy (I run Dingo,Puppeee,NOP)
In Antix I went to /etc/modprobe.d
I made a new blank text file called psmouse.conf
I typed into it
options psmouse proto=imps
Which basically tells my new kernel that I have a ps2 mouse instead of a pad mouse. Believe it or not this fixed the tap function for me.
Won't hurt to try it as you can delete the text file if it does not help.