More specifically, личка is a private message box where your private messages are kept. Написать в личку -- to send a private message

PM me your address. Личка — личное сообщение.

pm-private massage.It is generally used in online games

In terms of italki that means to give you an answer through a button “Message” allocated at a profile page. But in your case this button is unavailable that means I can`t sent this answer “в личку”. Harry

private message