To make the connection with Mitomaster working again on our server (redhat) I did two things:
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<li>
Add to the Perl script   : use LWP::Protocol::https;
<li>
Change the request to:  $request = POST 'https://mitomap.org/mitomaster/websrvc.cgi',
</ol>

To make the connection with Mitomaster working again in a virtual machine with CentOS6 I did three things:
<ol> 
<li>
Add to the Perl script: use LWP::Protocol::https;
<li>
Change the request to:  $request = POST 'https://mitomap.org/mitomaster/websrvc.cgi',
<li>
Install Perl packages perl-IO-Socket-SSL and perl-Net-SSLeay.
</ol>