I, personally, wouldn't dig deeper by completely reinstalling unless absolutely necessary. On the safe side I'd assume every file might erase, so you'd need to have that all backed up beforehand. As people suggested, it's probably a matter of using the proper relatively simple code and ensuring permissions. The suggested disk utility (careful what you use down there, though!) has worked for me as well when locked out of external drives, when I used verify disk. As for the path to the trash from the terminal, I just dragged a file into the terminal (it then supplies the path) and it looks like it's just .Trash/yourfile, and I noticed it also denied my permission and thought I was staff with reading privileges only.
I have a really stupid suggestion: can Mac download/auto-update you another mach file to the root of that computer? They auto-update other apps just b/c you're online. You'd have to contact them. Or, can they send you that file via any system/email/computer with full permissions so you can move it yourself no matter who the heck your computer thinks you are?
Otherwise, you might give that file writing permissions for everyone from the terminal (if disk utility disk verify doesn't give you back admin access). Then drag it or write where it should go in the terminal. And make it invisible if you wish. You could always change the permissions back to admin only with read for the others when you get it placed.
ls -l filenameWithFullPath tells you it's permissions with admin first, then group, then others...if I recall.
chmod 777 .Trash/mach_kernel should give it completely open permissions
mv .Trash/mach_kernel / should put it in the root
chmod 744 /mach_kernel will restore admin only privileges
sudo chflags hidden /mach_kernel should hide the file
If you didn't change the permissions on the trash file, use:
sudo mv .Trash/mach_kernel / should recognize you as admin with admin privileges to move the file
sudo chflags hidden /mach_kernel
More or less, if you have access to the terminal, something along these lines should be more or less correct. (On Windows the root is /. and I'm not doing this in real time with you, so there may be a slight discrepancy.). If there is an error probably someone will correct it, but the above won't do any damage.