How can I get my external hard drive to read and write?

Many brand-name drive-with-enclosures are shipped form the maker formatted for Windows New Technology File System (NTFS). Microsoft considers this format proprietary, and has said it will sue anyone who uses it without a license. Apple thinks paying money for proprietary File systems is nonsense, and will Read, but not Write that format without additional third-part software.

To use the drive as a Macintosh drive, you can completely ERASE it by its immutable device-name, not just its Volume, the entire device..

To continue to use it as a Windows drive, you would need to add a third-party driver. CAUTION: a Windows drive supported this way can only be read/written when the Windows NTFS Driver is loaded. That means NOT: Recovery, Installer or Safe Mode. You are also unlikely to be able to Restore from this when used as a backup drive.

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A Time Machine backup drive should be dedicated to TM backups. If you want to also store other files there, partition the drive (or create two volumes if using APFS).