In some cases sshpass attempts find "assword" as the default password prompt indicator. But rsync can return similar string:
Enter passphrase for key '/home/user/.ssh/private_user_key':
So, try to add '-P' parameter:
sshpass -p "yourpassword" -P 'Enter passphrase for key' rsync 111.111.111.111:/backup/origin /backup/destination/
Path to your private key you can set in /home/user/config or set with -e parameter like that:
sshpass -p "yourpassword" -P 'Enter passphrase for key' rsync -e 'ssh -i /home/user/.ssh/private_user_key' 111.111.111.111:/backup/origin /backup/destination/
More inf about default password prompt indicator:
$ sshpass -V
sshpass 1.06
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(C) 2015-2016 Shachar Shemesh
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Using "assword" as the default password prompt indicator.