Your problem is that you send the filename and the content of the file without any marker in between. Without the sleep both the name and the content might be send in a single packet, especially if the file is small. Assume for instance you have a 50 byte file, and a 20 byte filename. You send 70 bytes, and those 70 bytes will be read at the receiving end at your first sysread. By the time you get to the second sysread, in the loop, the sending end has already finished sending the data and closed the socket.
Your problem is that you are mixing protocols. It seems you like to do a line based protocol, first sending the filename, then the content, line-by-line, but you are using
a bytestream mechanism for it. If you want to do it line-by-line, just use print (on the sending end) and <$socket> (on the receiving end).