I have a friend who made a video of a spoon, but something was in the reflection. Is there a way to clean up, enhance, etc. to see it clear?

What he saw was a reflection of himself and no one else.

All the visual information is already in the video which you see. It may be possible to enhance the contrast to get slightly better rendering of part of the spoon.

Always work on a COPY FILE! 

A static spoon seems a strange thing to make a video of. If you want simply a still photograph, then take a still photograph, the quality will be better. Use soft light for highly reflective things such as spoons.

"Is there a way to clean up, enhance, etc. to see it clear?"

Not in the Hollywood sense that you adjust a slider and everything's perfect!  You could take a still from the video and play with the warp tools in an image editor.

Each individual frame would have to be edited. 

Does that sound like something you could do, 

or does it make re-shooting the video sound like a better idea?

i would ask your friend what it was