By default, an ext4 partition will have its root folder be owned by root. You can bypass this with certain mount options, but you should probably go and edit the owner of the root directory for your new drive. You can either change the owner to your normal user account, or change the permissions to permit any user of any group to read, write, and browse (also called “execute”) the directory. In most file managers, you can do this by right clicking an empty spot within the mounted directory and clicking “properties”. If you’re using the command line, you’re probably looking for chown $USER:$USER /path/to/mounted/drive/ or chmod 777 /path/to/mounted/drive.

There’s also a special “sticky bit” that directories can have in their permissions that make it difficult to write files to them, though I don’t think this applies in your case.

If you have done this already, make sure the disk is mounted somewhere your normal programs can access. If you mount it in /some/directory/drive, programs will throw up errors if they don’t have permissions to access /some or /some/directory.

If none of this works, we’ll need more information to help you. What mount options do you use, what directory are you mounting it in, who’s the current owner of the drive, what are the permissions like at the moment, what programs are failing to write, etc.