Welcome to the forum @Andrea_C!
Sounds like you've identified the source of the error, as well as a fix (using cutadapt to remove empty sequences), let me see if I can answer your remaining concerns:
Sometimes sequencing cores/services will run preliminary QC to filter low-quality sequences. You could discuss with them to see if/what they do, and maybe even get a rawer form of your data 
I am not 100% sure, but I don't think it will cause issues downstream if you just remove those sequences (or so that seems to be the case with others who have had similar issues, for example).
So you can use cutadapt to remove those sequences (and if there's an option to remove the paired sequences of these empty seqs that would be best just to avoid possible hitches downstream).
Give that a try and let us know what happens! If you are able to complete demultiplexing and denoising/clustering without error then you should be in the clear... it will have no downstream consequences beyond that stage.