I don't know of a way to get back the filename from a filehandle, and i think it's because a filehandle can be also a pipe or something else - not just a file-handle.
but the error you're getting is telling me that you probably should test if your object exists before you're calling the method "blahblah" on it (i can just guess as you haven't shown any code). then if the object does not exist you can add the appropriate exception handling.
for knowing which file you are processing i would probably just store the current filename in a variable.
if that's not what you meant please elaborate and show us some (few) lines of code.