Can't erase files

When I try to delete shortcuts on the desktop they go to
the recycle bin. Then when I try to empty the bin they
reappear on the desktop but with an additional copy. As
long as I don't empty the bin the desktop looks fine BUT,
when you explore the desktop the files are there; I have
28 copies of some; only visible while exploring, not
while normally using the desktop.

The next time I do empty the recycle bin all these
shortcuts will reappear on the desktop with a new copy of
each one. That is going to be about 187 shortcuts that I
then need to put into the recycle bin to hide.

Anyone have any ideas of how to fix this?

Thanks,

Jim