ok, i will try
Searching the internet I found some kernels. You may google for patwood kernel
3.4.67 is mine, but there are even newer versions.
If you dont find these you may try romanrm kernel for a10 too.
Then, what do you need to do:
First, replace the actual kernels - its the uImage file that you find in the first partition
Second, in the kernel zip file, you will see there are many modules, these are files with .ko extension.... locate the file, maybe copy them to your tinycore sd card, and then use insmod, for instance
sudo insmod 8188eu.ko
(for my wifi driver who didnt exist in the tinycore install)
by the way, when you switch this kernel, the first module you have to locate,
is named squashfs.ko
if you dont insert it the whole tinycore system will fail to work.
I hope this was more specific and possibly understandable, but i know it looks strange - it was difficult for me too