iPhone can't backup with iTunes because an error occurred while reading from or writing

Read/write usually happens when permissions don't align. Check your computer for backups of that phone that have different permissions. iTunes stores the backup files in the following locations:

Mac : ~/Library/Application Support/MobileSync/Backup/

Windows 7 and later: \Users\(username)\AppData\Roaming\Apple Computer\MobileSync\Backup\

Locate your iPhone backup files and check the permissions for the folder. Make the Privilege is "Read & Write". And the your current backup may be a corruption. If you need those backup files, you can move these files to another location on your computer or a USB device. Then go to iTunes -> Preferences -> Devices -> Device backups and there will be a button to "Delete Backup. Delete the current back up and try again to see if the issue is fixed.

Then:

1. Restart both your iPhone and a computer
2. Make sure iTunes is updated
3. Try another USB port
4. Make sure that anti-virus programs are disabled.

Try to back up again after every step.