I have a small issue with a bearing cover/seal (not sure how it's called properly) that I would like to fix. I'm attaching the picture for clarity. I imagine I should just replace the part but I have no idea how to go about looking for it. It's an old Bianchi racing bike.
You have the sort of headset where the adjustable race consists of a plastic shield fixed to a steel piece that has the bearing surface and the wrench flats. I think it might be a Shimano Exage Sport.
If you have access to the sort of used bike/parts shop that has bins of old stray headset parts, you might be able to find a replacement. But what you've got has never been super common and even then it's only one model, so you might be looking for a while. On the plus side, uppers tend to die last, so you might be able to find an orphaned one someone squirreled away.
Printing a replacement would be pretty reasonable here too, if you're into that kind of thing. You could saw it in half through the the middle to get the cross section and then lay that on a scanner, and from there make up a new plastic seal part in Fusion 360 or whatever you like and snap it onto the existing bearing race part.
Probably the most reasonable thing is to just replace the headset, or even just the upper.
I'm afraid the upper race of the upper bearing in your headset is irreversibly damaged. You need to replace it since ducktape is not the best material to be used as bearing race.
First of all you need to check which headset you have - the make (if you want to stick to the original), the threading (it can either be English/BSC or Italian, the latter is more likely since the bike is Bianci). What's almost sure is that it is a 1" headset.
When you go searching for the replacement part you most likely find the complete headset, unless LSB or some other hobbyist has and incomplete headset and can share the part you're missing.
What you need to get is the middle part in the top row in this picture (found on ebay):
And since this part is threaded, it is crucial to match the thread on your fork's steerer tube.