Mount a HFSPlus disk with read/write permissions in Linux

As we found out in the comments, there are two possible problems at hand here:

1. You are trying to run the copy as a regular user. This may cause reading the source files to fail because you don't have read permissions to them. This is easily corrected by running the copying through sudo just like you did the mount.
2. You get cp: omitting directory Documents/ when trying to run the copying through sudo. That's not a permissions problem at all, and can be fixed by simply telling cp to include subdirectories.

Putting these two together, you should be able to copy your files using a command like sudo cp -av /media/myharddrive /somewhere/else, where /somewhere/else exists and is writable.

The -v parameter isn't strictly needed, but after half an hour or an hour of just waiting, you might appreciate the files being listed as they are being copied. Note that if you have a very large number of small files, the screen refreshes may reduce the throughput of the copying; in that case, simply minimize the window and check on it occasionally.

-a tells cp to operate in "archive" mode, preserving as much as possible about the files it is copying, including subdirectories. Or you could use -r to tell it to only preserve the directory structure.

Using this, you should be able to copy the files to a more suitable location where you can work with them more freely, without being restricted by the read-only limitation of HFS+ file system support.