already in catalog

There's an apparent contradiction here, that needs to be reconciled - my own comments are indicated [thus]:

"It shows the copied photo's in LR [I assume, in the import screen] but makes them grey and tells me that they are already imported. If I click on those pictures in LR [which means they are visible inside the Catalog], and ask to show them in the explorer, it tells me that they are on the main drive in the right directory. But they still do not show in LR [you have just stated you were able to click on them though]. "

One explanation that would resolve that apparent contradiction, would be if edited images are appearing in the Catalog OK (for example, findable by a search or continuing to have membership in a Collection) but are not showing up when you go to their expected location within the Folders panel. 

There are two important tools for troubleshooting this kind of situation, accessed by right-clicking on a given image thumbnail that (as above), is findable inside LrC. The first you have already tried: "Show in Explorer" / "Show in Finder". The other informative test is: "Go to Folder in Library". Since the images are not showing up where expected, this has to take you somewhere unexpected - by definition. It must take you somewhere since that's what being imported in the Catalog means.

That 'somewhere' might correspond to a currently available drive and folder path, or it might not. Just as, when we try calling a 'wrong' phone number, that might ring an unexpected person, or we might just get "number unavailable".

Either way, this is fixed by "re-addressing" the Catalog's knowledge of where these images live - correcting the 'wrong' phone number to a 'right' phone number. Sometimes this is a situation dependent mistake, and a common example is that the Catalog and the Operating System treat variation in capital / lowercase letters within folder names, differently behind the scenes. The "Go to Folder in Library" tool troubleshoots what the actual situation is, and will direct what resolution is needed, which may sometimes be non-obvious on the face of it.

Under the same analogy, a person's phone number may incude the international country prefix or it may not. When you call that person from within their own country, either option will work equally well. When you call that person from overseas, only one of these two options will succeed. 

That resolution will never involve removing edited images from the Catalog, and it will never involve re-importing images. However if you did succeed in re-importing images again as unedited duplicates, that would block a resolution until this doubled importation had been undone.

When LrC greys out images and stops you from importing them, that is generally it being actively helpful - not obstructive. There are some rare instances where spurious matches do occur however - where LrC is in fact wrong that particular images are already present. This may be due to corrupted preferences, a corrupted Catalog database, or some externality. Whether that warning was spurious OR correct, ignoring it and forcing the images to import regardless would be to still leave unresolved whatever was the cause. It can then be expected to happen again at some future point.

If the folder and photos are in your LrC catalog but missing the photos, there will be a ? or ! .  I believe if you right-click the folder/photo you will be given the option to locate the folder/photo.  Go to the highest level folder with a ? or !, and and locate that filder. LrC will find the other folders/photos that are in sub-filders.

You can also go the the Library module, then pull down the Library menu and select "Find All Missing Photos".

Do not re-import them since you will lose the edits you have already done.

So I copied the original photos from the backup to the first drive (in the explorer) and wanted to import them in LR.

There is no importing here. Just use your operating system to copy the photos from the backup drive to the proper folder on the main drive.