If you can't put files onto Toshiba external hard drive from the MacBook Air, the reason may relate to the file permission and the drive permission. Besides, you can add files to an NTFS external drive from Mac using an NTFS driver for Mac without formatting. In case you want to format the drive, exFAT and FAT32 are compatible with both macOS and Windows.
First of all, check the permission for files you want to add to the external hard drive. Right-click the file and choose Get Info, check whether it is locked and whether the privilege is Read & Write. Besides, you can change permission on Mac for the external hard drive via Get Info > Share & Permissions, change it to Read - Write and tick Ignore the ownership of this volume to solve external hard drive read-only on Mac.
Then, make sure the drive format is supported by macOS such as APFS, HFS+, exFAT, and FAT32. In case the file format is NTFS, then you can only read but not write to it. If you have kept some important files on the NTFS drive, you may don't want to format it for Mac to lose all the data. Thus, NTFS for Mac software is what you need such as iBoysoft NTFS for Mac. It mounts the NTFS drive in read-write mode to let you access it without any limitations.
When you can't copy from Mac to external hard drive due to the incompatible file format, you can choose to format the drive directly if you don't mind data loss. To use the drive for both macOS and Windows, you should select the file format as exFAT or FAT32.
