How to connect and browse files of remote server via Midnight Commander's Shell link (copy files over SSH) if the private key requires password?

1. Create ~/.ssh/config file containing one or more entries along the lines of:

   Host your_server_name
    HostName 1.2.3.4
    Port 22
    User your_username
    IdentityFile ~/.ssh/your_private_key
   
   Host another_server
    HostName 4.5.6.7
    Port 22
    User another_username
   

   The private key for another server requires a password, so the IdentityFile line can be omitted as ssh-agent will be used:

2. Set the proper permissions on ~/.ssh/config:

   chmod 600 ~/.ssh/config
   

3. Start ssh-agent:

   eval "$(ssh-agent -s)"
   

4. Add private key with password:

   ssh-add ~/.ssh/private_key_for_another_server
   

5. In Midnight Commander, click

   Left or Right > Shell link...

   and enter "another_server" (or whatever name was given in ~/.ssh/config).

If everything looks right, but MC still refuses to connect, make sure that the private keys also have stricter permissions. On several occasions, I copied the private keys to a new "jump" server, but nothing happened, and just needed some chmod 600.

This answer is mostly a combination of:

1. the article Midnight Commander remote connect via Shell link (copy files over SSH) and SFTP link using FISH and public key authentication, and

2. this answer.

The article has lot of pictures and other examples, too.